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• Surface Area Of A Sphere

Surface Area of a Sphere

The surface area of a sphere is defined as the region covered by its outer surface in three-dimensional space.  A Sphere is a three-dimensional solid having a round shape, just like a circle. The formula of total surface area of a sphere in terms of pi ( π ) is given by:

The difference between a sphere and a circle is that a circle is a two-dimensional figure or a flat shape, whereas, a sphere is a three-dimensional shape. Therefore, the area of circle is different from  area of sphere. 

Area of circle = π r 2 

From a visual perspective, a sphere has a three-dimensional structure that forms by rotating a disc that is circular with one of the diagonals.

Let us consider an instance where spherical ball faces are painted. To paint the whole surface, the paint quantity required has to be known beforehand. Hence, the area of every face has to be known to calculate the paint quantity for painting the same. We define this term as the total surface area.

The surface area of a sphere is equal to the areas of the entire face surrounding it.

The surface area of a sphere formula is given  by,

A = 4 π r 2  square units

For any three-dimensional shapes, the area of the object can be categorised into three types. They are:

• Curved Surface Area
• Lateral Surface Area
• Total Surface Area

Curved Surface Area:   The curved surface area is the area of all the curved regions of the solid.

Lateral Surface Area : The lateral surface area is the area of all the regions except bases (i.e., top and bottom).

Total Surface Area:  The total surface area is the area of all the sides, top and bottom the solid object.

In case of a Sphere, it has no flat surface.

Therefore, the Total surface area of a sphere = Curved surface area of a sphere

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• Surface Area of a Sphere – Explanation & Examples

Surface Area of a Sphere – Explanation & Examples

The surface area of a sphere is the measure of the region covered by the surface of a sphere.

In this article, you will learn how to find the surface area of a sphere using the surface area of a sphere formula .

How to Find the Surface Area of a Sphere?

Like a circle, the distance from the center of a sphere to the surface is known as the radius. The surface area of a sphere is four times the area of the circle with the same radius.

Surface area of a sphere formula

The surface area of a sphere formula is given as:

Surface area of a sphere = 4πr 2  square units ……………. (Surface area of a sphere formula)

For a hemisphere (a half of a sphere), the surface area is given by;

Surface area of a hemisphere = ½ × surface area of sphere + area of the base (a circle)

= ½ × 4π r 2  + π r 2 

Surface of a hemisphere = 3πr 2 …………………. (Surface area of a hemisphere formula)

Where r = the radius of the given sphere.

Let’s solve a few example problems about the surface area of a sphere.

Calculate the surface area of a sphere of radius 14 cm.

Radius, r =14 cm

By the formula,

Surface area of a sphere = 4πr 2 

On substitution, we get,

SA = 4 x 3.14 x 14 x 14

= 2,461.76 cm 2 .

The diameter of a baseball is 18 cm. Find the surface area of the ball.

Diameter = 18 cm ⇒ radius = 18/2 = 9 cm

A baseball has a spherical shape, therefore,

The surface area = 4πr 2 

= 4 x 3.14 x 9 x 9

SA = 1,017.36 cm 2

The surface area of a spherical object is 379.94 m 2 . What is the radius of the object?

SA = 379.94 m 2

But, surface area of a sphere = 4πr 2 

⇒ 379.94 = 4 x 3.14 x r 2

⇒ 379.94 =12.56r 2

Divide both sides by 12.56 and then find the square of the result

⇒ 379.94/12.56 = r 2

⇒ 30.25 = r 2

⇒ r = √30.25

Therefore, the radius of the spherical solid is 5.5 m.

The cost of leather is $10 per square meter. Find the cost of manufacturing 1000 footballs of radius 0.12 m.

First, find the surface area of a ball

SA = 4πr 2 

= 4 x 3.14 x 0.12 x 0.12

= 0.181 m 2

The cost of manufacturing a ball = 0.181 m 2 x $10 per square meter

Therefore, the total cost of manufacturing 1000 balls = $1.81 x 1000

The radius of the Earth is said to be 6,371 km. What is the surface area of the Earth?

The Earth is a sphere.

= 4 x 3.14 x 6,371 x 6,371

= 5.098 x 10 8 km 2

Calculate the surface area of a solid hemisphere of radius 10 cm.

Radius, r = 10 cm

For a hemisphere, the surface area is given by:

Substitute.

SA = 3 x 3.14 x 10 x 10

So, the surface area of the sphere is 942 cm 2 .

The surface area of a solid hemispherical object is 150.86 ft 2 . What is the diameter of the hemisphere?

SA = 150.86 ft 2 .

Surface area of a sphere = 3πr 2

⇒ 150.86 = 3 x 3.14 x r 2

⇒ 150.86 = 9.42 r 2

Divide both sides by 9.42 to get,

⇒ 16.014 = r 2          

r = √16.014

Hence, the radius is 4 ft, but the diameter is twice the radius.

So, the diameter of the hemisphere is 8 ft.

Calculate the surface area of a sphere whose volume is 1,436.03 mm 3 .

Since, we already know that:

Volume of a sphere = 4/3 πr 3

1,436.03 = 4/3 x 3.14 x r 3

1,436.03 = 4.19 r 3

Divide both sides by 4.19

So, the radius of the sphere is 7 mm.

Now calculate the surface area of the sphere.

= 4 x 3.14 x 7 x 7

= 615.44 mm 2 .

Calculate the surface area of a globe of radius 3.2 m

Surface area of a sphere = 4π r 2 = 4π (3.2) 2 = 4 × 3.14 × 3.2 × 3.2 = 128.6 m 2

Hence, the surface area of the globe is 128.6 m 2 .

Practice Questions

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Surface Area of a Sphere

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• Mahindra Jain
• Jongheun Lee
• Geoff Pilling
• Yash Dev Lamba
• Gene Keun Chung
• Andres Gonzalez

A sphere is a perfectly round geometrical 3-dimensional object. It can be characterized as the set of all points located distance \(r\) (radius) away from a given point (center). It is perfectly symmetrical, and has no edges or vertices.

A sphere with radius \(r\) has a volume of \( \frac{4}{3} \pi r^3 \) and a surface area of \( 4 \pi r^2 \).

A sphere has several interesting properties, one of which is that, of all shapes with the same surface area, the sphere has the largest volume.

Archimedes' Hat-Box Theorem

Practice problems.

To prove that the surface area of a sphere of radius \(r\) is \(4 \pi r^2 \), one straightforward method we can use is calculus. We first have to realize that for a curve parameterized by \(x(t)\) and \(y(t\)), the arc length is \[ S = \int_a^b \sqrt{ \left(\frac{dy}{dt}\right)^2 + \left( \frac{dx}{dt}\right)^2 } \, dt. \] From this we can derive the formula for the surface area of the solid obtained by rotating this about the \(x\)-axis. This turns out to be \[ A = 2\pi \int_a^b y\sqrt{ \left(\frac{dy}{dt}\right)^2 + \left( \frac{dx}{dt}\right)^2 } \, dt .\] We can obtain a sphere by revolving half a circle about the \(x\)-axis. This circle can be parameterized as \(x(t)=r\cos(t) \) and \(y(t) = r\sin(t) \) for \(0 \leq t \leq \pi \). From this, we get \[ \frac{dx}{dt} = -r\sin(t), \quad \frac{dy}{dt} = r\cos(t) .\] Substituting in our equations for surface area gives \[\begin{align} A &= 2\pi \int_0^\pi r\sin(t)\sqrt{ \big(-r\sin(t)\big)^2 + \big( r\cos(t) \big)^2 } \ dt \\ &= 2\pi \int_0^\pi r\sin(t)\sqrt{ r^2\big(\sin(t)^2 + \cos(t)^2 \big) } \ dt \\ &= 2\pi \int_0^\pi r^2 \sin(t) \ dt \\ &= 2\pi r^2 \int_0^\pi\sin(t) \ dt \\ &= 4 \pi r^2. \ _\square \end{align} \]

Archimedes' Hat-Box Theorem Archimedes' hat-box theorem states that for any sphere section, its lateral surface will equal that of the cylinder with the same height as the section and the same radius of the sphere.

Let us recall our last proof section. After revolving the semicircle around the \(x\)-axis, we will obtain a sphere's surface area, and if we cut just a partial section with parallel bases, the new surface area will be demonstrated in the image below: From the image, the section's lateral surface area is colored light blue with 2 circular bases of different radii. In order to visualize the section's height better, this section will be rotated by 90 degrees, as shown below: Now inside the section, there are 2 variable angles, \(\angle a\) and \(\angle b\), which appear as the integral borders of the cut section. From the proof's conclusion, the surface area of the section \((A')\) can be calculated as \[\begin{align} A' &= 2\pi r^2 \int_a^b\sin(t) \ dt \\ &= 2\pi r^2 \left[\left.-\cos(t) \right|_a^b\right] \\ &= (2\pi r)r\big[\cos (a) - \cos (b)\big] . \end{align}\] Considering the right triangles with radius \(r\) (thick red) in the image, it is obvious that \(r\) is the hypotenuse side for both. As a result, the vertical sides can be calculated as \(r\times \cos (a)\) and \(r\times \cos (b)\) for the left and right triangles, respectively. Hence, the height of the section is \(h = \big(r\times \cos (a)\big) - \big(r\times \cos (b)\big) = r\big[\cos (a) - \cos (b)\big]\). Substituting this term to the previous equation gives \[A' = (2\pi r)r\big[\cos (a) - \cos (b)\big] = 2\pi rh. \] Clearly, this is the formula for the cylinder's lateral surface with radius \(r\) and height \(h\)! That means the lateral surface area of the sphere section equals the lateral surface area of the cylinder with radius \(r\) and height \(h,\) as shown in the image, and this holds true for any level of the sphere involved. \(_\square\)

A spherical tomato and a cylindrical portion of a cucumber have the same height and radius. Then they are chopped into slices of equal thickness, as shown above.

Comparing each slice of both kinds, which slice will have more lateral surface area of the peel?

A small green circle is inscribed within the section of a bigger blue circle, touching the mid-chord, as shown above left. Then the graphs are revolved around the \(y\)-axis to generate three figures: a blue cover dome, a green spherical melon, and a red serving plate.

Which of the following options will have more surface area?

I. The blue dome II. The melon plus the plate

A sweets shop sells candies in 2 different styles: a spherical ball and a dome. The dome-like shape is a spherical section of a larger sphere with height \(h\) and base radius \(R,\) as shown above, while the candy ball has radius \(r\) with \(2r = R + h\).

If both shapes have the same total surface area, what is the ratio \(\frac{R}{h}\)?

What is the surface area of a sphere of radius 3? The surface area is \( 4 \pi \times 3^2 = 36 \pi \). \( _\square \)

If the volume of a sphere is \(36\pi,\) what is the surface area of the sphere? Observe that the volume of the sphere can be rewritten as \[36\pi=\frac{4}{3}\pi \times 3^3.\] Then, since the volume of a sphere with radius \(r\) is \( \frac{4}{3} \pi r^3 ,\) it follows that the radius of the sphere in this problem is \(r=3.\) Hence, its the surface area is \[4 \pi r^2 =4\pi \times 3^2 =36\pi. \ _\square\]

The volume of a sphere has grown 8 times. Then how many times has the surface area grown in the meanwhile? Observe that the volume of the sphere is \( \frac{4}{3} \pi r^3.\) This implies that it is proportional to \(r^3,\) that is\( \frac{4}{3} \pi r^3 \propto r^3\). Then 8 times growth in the volume of the sphere implies 2 times growth in the radius of sphere. Then, since the surface area of sphere is \( 4 \pi r^2 \propto r^2, \) the surface area of the sphere has grown \(2^2 = 4\) times. \(\ _\square\)

You have a watermelon whose volume is \(288 \text{ cm}^3.\) If you cut the watermelon into halves, what is the surface area of one half of the watermelon? (Assume that the watermelon is a perfect sphere.) From the formula \( V=\frac{4}{3} \pi r^3 \) for the volume of a sphere with radius \(r,\) you know that the radius of the watermelon is \(r=6 \text{ cm}.\) Since you cut the watermelon into two exact halves, you may think that the surface area of a half watermelon is also exactly half the surface area of the whole watermelon. However, this thinking is wrong. As shown int the above diagram, the surface area of a half watermelon is bigger than half the surface area of a whole watermelon, by the area the cross section \(A.\) Thus, the surface area of a half watermelon is \[\text{(Half the surface area of the watermelon)} + \text{(Area of A)}. \] Since \(A\) is a circle whose radius is the same as the radius of the watermelon, our answer is \[\frac{1}{2} \times 4\pi \times 6^2 + \pi \times 6^2 = 108 \pi. \ _\square\]

The diameter of a solid metallic right circular cylinder is equal to its height. After cutting out the largest possible solid sphere \(S\) from this cylinder, the remaining material is recast to form a solid sphere \(Q.\)

What is the ratio of the radius of sphere \(S\) to that of sphere \(Q?\)

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Surface area of a sphere

Here you will learn about the surface area of a sphere, including how to calculate the surface area of a sphere given its radius and the surface area of a hemisphere.

Students will first learn about the surface area of a sphere as a part of geometry in 8 th grade.

What is the surface area of a sphere?

The surface area of a sphere is the area which covers the outer surface of the sphere. It includes both the curved surface area of a sphere and the area of the sphere’s circular base, also known as the total surface area of a sphere ( tsa ).

The formula for the surface area of a sphere with radius r is:

\text{Surface area}=4\pi{r}^{2}

Notice that the square of the radius (r^2) occurs within the surface area formula.

The surface area of a shape uses two-dimensions , so the units for surface area are units squared.

Common Core State Standards

How does this relate to 8 th grade math?

• Grade 8: Geometry (8.G.C.9) Know the formulas for the volumes of cones, cylinders, and spheres and use them to solve real-world and mathematical problems.

[FREE] Surface Area Worksheet (Grade 6)

Use this quiz to check your grade 6 students’ understanding of surface area. 10+ questions with answers covering a range of 6th grade surface area topics to identify areas of strength and support!

How to calculate the surface area of a sphere

In order to calculate the surface area of a sphere:

Write down the formula.

Substitute the given values into the formula.

Complete the calculation.

Write the final answer, including the units.

Surface area of a sphere examples

Example 1: surface area of a sphere given the radius.

Find the surface area of the sphere below. Write your answer to 1 decimal place.

To answer the question, use the formula for the surface area of a sphere:

2 Substitute the given values into the formula.

The radius (the distance from the center of the sphere to the surface) is 4.8 \, cm.

Substitute the value of the radius into the formula for the surface area of a sphere:

3 Complete the calculation.

Use a calculator to find the value for the surface area:

4 Write the final answer, including the units.

Check what form the final answer needs to be. Here you are asked to give the answer to 1 decimal place.

The surface area of the sphere is 289.5 \mathrm{~cm}^2 \, \text{(1dp)}.

Example 2: surface area of a sphere

Find the surface area of a sphere with the diameter of 42 \mathrm{~mm}.

Write your answer to the nearest square millimeter.

\text{Surface area}=4 \pi r^2

Because the diameter is twice the length of the radius, divide the diameter by 2 to get the value for r \text{:}

\begin{aligned}r&=d\div{2} \\\\ &=42\div{2} \\\\ &=21\end{aligned}

So the radius r=21 \, mm. Now, substitute r=21 into the formula for the surface area of a sphere to get:

\text{SA }=4 \times \pi \times 21^2

Use a calculator to work out the value.

\begin{aligned}\text{SA }&=4 \times \pi \times 21^2 \\\\ &=1764\pi \\\\ &=5541.769441… \end{aligned}

Here you are asked to write the answer to the nearest square millimeter and so:

5541.769441… = 5542 \, \text{mm}^2 \, \text{(0dp)}

The surface area of the sphere is 5542 \mathrm{~mm}^2 \, \text{(0dp)}.

Example 3: surface area of a sphere – in terms of π

Find the surface area of a sphere with a radius of 6 \, m.

Leave your answer in terms of \pi.

Substitute the value of the radius into the surface area of a sphere formula.

\text{SA }=4 \times \pi \times 6^2

Work out the curved surface area, focussing on the number parts of the calculation.

Here you are asked to give the answer in terms of \pi.

144\pi\text{ m}^2

The surface area of the sphere is 144\pi\text{ m}^2.

Example 4: surface area of a sphere

Find the surface area of the sphere below. Write your answer to 2 decimal places.

The radius (the distance from the center of the sphere to the surface) is 12.4 \, cm.

\text{SA }=4 \times \pi \times 12.4^2

\begin{aligned}\text{SA }&=4 \times \pi \times 12.4^2 \\\\ & =615.04 \pi \\\\ & =1932.20515 \ldots\end{aligned}

Check what form the final answer needs to be. Here you are asked to give the answer to 2 decimal places.

1932.20515 . \ldots=1932.21 \mathrm{~cm}^2 \, \text{(2dp)}

The surface area of the sphere is 1932.21 \mathrm{~cm}^2 \, \text{(2dp)}.

The surface area of a sphere is 600 \mathrm{~cm}^2. Calculate the radius of the sphere. Write your answer to 1 decimal place.

The surface area is given, so substitute this into the formula.

600=4 \pi r^2

Rearrange the formula to find the value of r.

The radius of the sphere is 6.9 \mathrm{~cm} \, \text{(1dp)}.

The surface area of a sphere is 4000 \mathrm{~cm}^2. Calculate the radius of the sphere. Write your answer to 1 decimal place.

4000=4 \pi r^2

The radius of the sphere is 17.8 \mathrm{~cm} \, \text{(1dp)} .

Teaching tips for surface area of a sphere

• Use visuals, such as diagrams and real life examples, to help students visualize what the shape of a sphere looks like and what it means to calculate the surface area.
• Provide practice problems at various difficulty levels for students to work through. This can be done with worksheets, or with interactive technology. Allow students to work together and independently to reinforce understanding.
• Have students find the surface area of multiple shapes and compare them. This will better help students understand the formulas they use, and why certain shapes are used in different real life situations.

Easy mistakes to make

• Rounding too soon It is important to not round the answer until the end of the calculation. This will mean your final answer is accurate.
• Not using the correct units For area, use square units such as \mathrm{~cm}^2. For volume, use cube units such as \mathrm{~cm}^3.
• Incorrectly using the radius or the diameter It is a common error to mix up radius and diameter. Remember the radius is half of the diameter.
• Confusing the volume formula with the surface area formula It is a common error to mix up the formula for the volume of a sphere with the formula for the surface area of a sphere. Remember that the surface area formula includes a squared term, whereas the volume formula includes a cubed term.

Related surface area lessons

• Surface area of a prism
• Surface area of a rectangular prism
• Surface area of a triangular prism
• Surface area of a pyramid
• Surface area of a cone
• Surface area of a cylinder
• Surface area of a cube
• Surface area of a hemisphere

Practice surface area of a sphere questions

1. Find the surface area of a sphere of radius 10 \, cm. Give your answer to 1 decimal place.

You are finding the surface area of a sphere, so substitute the value of r into the formula.

2. Find the surface area of a sphere of diameter 6.4 \, cm. Write your answer to 1 \, dp.

To find the surface area of a sphere, substitute the value of r into the formula. The diameter is given, so half 6.4 to get the radius, then substitute this value into the formula for the surface area of a sphere.

3. Calculate the surface area of a sphere of radius 9 \, m.

To find the surface area of a sphere, substitute the value of r into the formula.

4. Calculate the surface area of the sphere. Write your answer to 2 decimal places.

5. The surface area of a sphere is 9000 \mathrm{~cm}^2. Calculate the radius of the sphere. Write your answer correct to 3 significant figures.

Using the formula, we substitute the value of the surface area and rearrange to find the radius.

6. A spherical ball has a radius of 30 \, cm.

A tin of paint covers 2000 \mathrm{~in}^2.

A tin of paint costs \$ 13.

Calculate how much it would cost to paint the outside surface of the spherical ball.

To find the surface area of a sphere, substitute the value of r, the radius of the ball, into the formula.

Next, divide the surface area by the number of square inches that a tin of paint covers.

Multiply the number of tins needed by the total cost of each tin of paint.

Surface area of a sphere FAQS

A sphere is a three-dimensional shape that is perfectly round. It can be defined as the set of all points in three dimensional space that are a fixed distance (radius) from a given point (the center).

The diameter of a sphere is twice the length of its radius. If you are given the radius of the sphere, simply double it to find the diameter. Alternatively, if you are given the diameter and need the radius, you can divide the diameter by 2 to find the radius.

A circle is a two-dimensional shape with a flat surface. A sphere is a three dimensional shape that is perfectly round, like a ball. The circle can also be thought of as the cross section of a sphere.

Did you know?

Archimedes was a famous ancient Greek mathematician who lived about 2,200 years ago in Sicily. He wrote about the surface area of a sphere and said it was four times the area of the greatest circle. This is the same formula that is used today.

Archimedes also worked on cylinders and discovered the formula for the lateral surface area of a cylinder. The lateral surface area excludes its circular base and top. This is usually referred to as its curved surface area.

The next lessons are

• Pythagorean Theorem
• Trigonometry
• Circle math

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Surface Area of Sphere

The surface area of a sphere is the area occupied by the curved surface of the sphere. Circular shapes take the shape of a sphere when observed as three-dimensional structures. For example, a globe or a soccer ball. Let us learn about the formula of surface area of a sphere and how to calculate the surface area of a sphere in this lesson.

What is the Surface Area of a Sphere?

The area covered by the outer surface of the sphere is known as the surface area of a sphere. A sphere is a three-dimensional form of a circle . The difference between a sphere and a circle is that a circle is a 2-dimensional shape ( 2D shape ), whereas a sphere is a 3-dimensional shape. The surface area of a sphere is expressed in square units. Observe the sphere given below which shows the center, the radius, and the diameter of a sphere.

Sphere Definition

Sphere is a three-dimensional round-shaped object with no vertices or edges. The important aspects of this shape are radius, diameter, circumference , and volume.

Derivation of Surface Area of Sphere

A sphere is round in shape, therefore to find its surface area, we relate it to a curved shape, such as the cylinder. A cylinder is a shape that has a curved surface along with flat surfaces. Now, if the radius of a cylinder is the same as the radius of a sphere, it means that the sphere can fit into the cylinder perfectly. This means that the height of the cylinder is equal to the height of the sphere. So, this height can also be called as the diameter of the sphere. Therefore, this fact was proved by a great mathematician, Archimedes, that if the radius of a cylinder and sphere is 'r', the surface area of a sphere is equal to the lateral surface area of the cylinder .

Hence, the relation between the surface area of a sphere and lateral surface area of a cylinder is given as:

Surface Area of Sphere = Lateral Surface Area of Cylinder

The lateral surface area of a cylinder = 2πrh, where 'r' is the radius and 'h' is the height of the cylinder. Now, the height of the cylinder can also be called the diameter of the sphere because we are assuming that this sphere is perfectly fit in the cylinder. Hence, it can be said that height of the cylinder = diameter of sphere = 2r. So, in the formula, surface area of Sphere = 2πrh; 'h' can be replaced by the diameter, that is, 2r. Hence, surface area of sphere is 2πrh = 2πr(2r) = 4πr 2

Formula of Surface Area of Sphere

The formula of the surface area of the sphere depends on the radius of the sphere. If the radius of the sphere is r and the surface area of the sphere is S. Then, the surface area of the sphere is expressed as:

Surface Area of Sphere = 4πr 2 ; where 'r' is the radius of the sphere.

In terms of diameter, the surface area of a sphere is expressed as S = 4π(d/2) 2 where d is the diameter of the sphere.

How to Calculate the Surface Area of Sphere?

The surface area of a sphere is the space occupied by its surface. The surface area of the sphere can be calculated using the formula of the surface area of the sphere. The steps to calculate the surface area of a sphere are given below.

Let us take an example to learn how to calculate the surface area of a sphere using its formula.

Example: Find the surface area of a spherical ball that has a radius of 9 inches.

• Step 1: Note the radius of the sphere. Here, the radius of the ball is 9 inches.
• Step 2: As we know, the surface area of sphere = 4πr 2 , so after substituting the value of r = 9, we get, surface area of sphere = 4πr 2 = 4 × 3.14 × 9 2 = 4 × 3.14 × 81 = 1017.36
• Step 3: Therefore, the surface area of the sphere is 1017.36 in 2

Curved Surface Area of Sphere

The curved surface area of a sphere is the total surface area of the sphere because a sphere has just one surface which is curved. Since there is no flat surface in a sphere, the curved surface area of a sphere is equal to the total surface area of the sphere. Therefore, the formula for the curved surface area of a sphere is expressed as, Curved surface area of sphere = 4πr 2 ; where 'r' is the radius of the sphere.

☛ Related Articles

• Surface Area of Cube
• Surface Area of Cuboid
• Surface Area of Prism
• Surface Area of Cone
• Difference Between Area and Surface Area

Surface Area of Sphere Examples

Example 1: If the radius of a sphere is 20 feet, find its surface area. (Use π = 3.14). Solution: Given, the radius 'r' of the sphere = 20 feet.

The surface area of the sphere = 4πr 2 = 4 × π × 20 2 = 5024 feet 2

∴ The surface area of the sphere is 5024 feet 2

Example 2: Find the surface area of a sphere if its radius is given as 6 units.

Solution: Given, the radius 'r' = 6 units. So, let us substitute the value of r = 6 units

⇒ The surface area of the sphere = 4πr 2 = 4 × π × 6 2 = 4 × 3.14 × 36 = 452.16 unit 2

∴ The surface area of the sphere is 452.16 unit 2

Example 3: State true or false.

a.) A sphere is a three-dimensional form of a circle.

b.) The curved surface area of a sphere is the total surface area of the sphere because a sphere has just one surface which is curved.

a.) True, a sphere is a three-dimensional form of a circle.

b.) True, the curved surface area of a sphere is the total surface area of the sphere because a sphere has just one surface which is curved.

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Practice Questions on Surface Area of Sphere

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FAQs on Surface Area of Sphere

What is the surface area of sphere in math.

The surface area of a sphere is the total area that is covered by its outer surface. The surface area of a sphere is always expressed in square units. The formula for the surface area of a sphere depends on the radius and the diameter of the sphere. It is mathematically expressed as 4πr 2 ; where 'r' is the radius of the sphere.

Why is the Surface Area of a Sphere 4 Times the Area of a Circle?

A string that completely covers the surface area of a sphere can completely cover the area surface of exactly four circles. This way you can check that the surface area of a sphere is four times the area of a circle. When we write the formula for the surface area of a sphere, we write the surface area of a sphere = 4πr 2 = 4(πr 2 ) = 4 × area of a circle.

How Many Sides and Vertices Does a Sphere Have?

A sphere is a three-dimensional shape that is round like a circle. Hence, it has no sides, vertices, or faces.

Does a Sphere have Infinite Faces?

No, a sphere has no face. A face is a flat surface and a sphere has no flat surface. This makes the sphere a faceless three-dimensional shape ( 3D shape ).

What is the Curved Surface Area and Total Surface Area of a Sphere?

A sphere has just one surface and that is curved. Since there is no flat surface in a sphere, the curved surface area of a sphere is equal to its total surface area of the sphere which is 4πr 2 .

☛Also Check:

• Surface Area Formulas
• Geometry Formulas
• Measurement Formulas

What is the Surface Area of a Sphere Formula in Terms of Diameter?

The surface area of a sphere formula in terms of diameter is given as, πD 2 where 'D' is the diameter of the sphere. It gives the relationship between the surface area of a sphere and the diameter of the sphere.

How to Calculate the Surface Area of a Sphere With the Volume?

The surface area of a sphere can be easily calculated with the help of the volume of the sphere. In this case, we should know the value of the radius of the sphere. The radius of the sphere can be calculated from the formula of the volume of the sphere, that is, Volume of a sphere = 4/3 × πr 3 . From this, the radius can be calculated and then its value is substituted in the formula for the surface area. We know that the surface area of the sphere = 4πr 2 . Another way to follow this is as follows. From the volume formula, we can derive that, r 3 = 3V/4π, or r = (3V/4π) 1/3 . After this, we can substitute the value of r in the surface area of the sphere formula.

What is the Surface Area of Sphere Calculator?

Surface area of sphere calculator is an online tool available for kids to ease their calculations. It is system generated tool where the surface area formula is pre-fixed all we have to do is enter the value of the given parameters, such as radius and we get the surface area of the sphere. Try now Cuemath's surface area of a sphere calculator and get your answers in a few seconds.

How Does the Surface Area of Sphere Change When the Radius is Halved?

The surface area of the sphere gets one-fourth when the radius is halved because r becomes r/2. As, the surface area of a sphere = 4πr 2 , so, if we replace 'r' with r/2, the formula becomes 4π(r/2) 2 = πr 2 which is one-fourth of the surface area. Thus, the surface area of the sphere gets one-fourth as soon as its radius gets halved.

How does the Surface Area of a Sphere Change When the Radius is Tripled?

The surface area of the sphere becomes 36πr 2 when the radius is tripled because 'r' becomes 3r'. We know that the surface area of a sphere = 4πr 2 , so if we replace 'r' with 3r, we get the formula as, surface area = 4π(3r) 2 = 4π × 9r 2 = 36πr 2

• Surface ...

Surface Area of a Sphere

In this tutorial, we'll learn how to find the surface area of a sphere. And given how often we see the spherical shape around us, I am sure we all have some intuitive understanding of what a sphere is.

So, a spherical surface is the set of all points in space that are at a fixed distance (radius, usually denoted by r \hspace{0.2em} r \hspace{0.2em} r ) from a given point (center, O \hspace{0.2em} O \hspace{0.2em} O ).

It’s very similar to the concept of a circle. The key difference is – a circle is the set of all points in a plane (instead of space) that are equidistant from a given point.

Unlike a cone, cube, or cylinder, a sphere does not have any edges. So a sphere has only one continuous surface. And the area covered by this surface is the surface area of the sphere.

Formula | Surface Area of a Sphere

For a sphere with a radius r \hspace{0.2em} r \hspace{0.2em} r , the surface area is given by

Archimedes, the famous Greek polymath, found that the surface area of a sphere is equal to the curved surface area of a cylinder with a radius equal to the sphere's radius and height equal to the sphere's diameter.

Now, the curved surface area of a cylinder is given by –

Replacing with radius r \hspace{0.2em} r \hspace{0.2em} r and height 2 r \hspace{0.2em} 2r \hspace{0.2em} 2 r is given by

Hence the curved surface area of a sphere must be equal to

How to Find the Surface Area of a Sphere | Examples

Alright. Time to solve a few examples using what we have learned so far.

Find the surface area of a sphere with a diameter of 28 \hspace{0.2em} 28 28 inches.

The surface area of a sphere is given by -

But the question doesn't give us the radius ( r ) \hspace{0.2em} (r) ( r ) . Instead, it tells us the diameter is 28 \hspace{0.2em} 28 \hspace{0.2em} 28 inches. So first we need to get the radius.

Now, substituting the value of the radius into our formula for area, we have

So the surface area of the sphere is 2463  in 2 \hspace{0.2em} 2463 \text{ in}^2 \hspace{0.2em} 2463  in 2 .

What is the radius of a sphere that has a surface area of 36 π \hspace{0.2em} 36 \pi \hspace{0.2em} 36 π ?

Again, the surface area of a sphere is given by –

And the question tells us that the area is 36 π \hspace{0.2em} 36 \pi \hspace{0.2em} 36 π . So,

Solving for r \hspace{0.2em} r \hspace{0.2em} r , we get

And with that we come to the end of this tutorial. Until next time.

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Surface Area of a Sphere

Learn how to find the surface area of a sphere., surface area of a sphere lesson, sphere surface area formula.

The formula for surface area of a sphere is given as:

Where SA is the surface area and r is the radius of the sphere.

Surface Area of a Sphere Example Problems

Let's go through a couple of example problems together to practice finding the surface area of a sphere.

Example Problem 1

Find the surface area of a sphere with a radius of 10.

• Plugging what we know into the surface area formula, we get:
• SA = 4π(10 2 ) = 400π
• The surface area is 400π.

Example Problem 2

The surface area of a sphere is measured to be 100π square meters. What is the diameter of the sphere in meters?

• First, we will plug the surface area value into the formula and solve for the radius r .
• 100π = 4πr 2
• Now that we have the radius, let's convert it to diameter.
• d = 2(5) = 10
• The diameter of the sphere is 10 meters.

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Last modified on April 25th, 2024

#ezw_tco-2 .ez-toc-title{ font-size: 120%; ; ; } #ezw_tco-2 .ez-toc-widget-container ul.ez-toc-list li.active{ background-color: #ededed; } chapter outline

Surface area of a sphere.

The surface area of a sphere is the entire region covered by its outer round surface. It is also the curved surface area of a sphere. Like all other surface area it is expressed in square units such as m 2 , cm 2 , and mm 2 .

We will learn how to find the surface area of a solid sphere. The equations are given below.

The basic formula is given below:

With Radius

Let us derive the formula of surface area.

According to Archimedes, if a sphere and a cylinder have equal radius, ‘ r ’, then,

Surface area of a sphere = Lateral surface area of a cylinder

Lateral surface area of the cylinder = 2πrh, here r = radius, h = height

Assuming the sphere perfectly fit within the cylinder

Height ( h ) of the cylinder = diameter of the sphere = 2r …………. (1)

∴Lateral surface area of the cylinder = 2πrh

= 2πr × (2r) …………from (1), ∵ h = 2r

Thus the surface area of a sphere = 4πr 2

Let us solve an example involving the above formula.

Find the surface area of a sphere whose radius is 5 in.

As we know, Surface Area ( SA ) = 4πr 2 , here π = 22/7 = 3.141, r = 5 in ∴ SA = 4 × 3.141 × 5 2 = 314.1 in 2

Let us find the surface area of a sphere when the radius is not given directly.

With Diameter

The formula to find the surface area of a sphere using diameter is:

Find the surface area of a sphere with a radius of 9 cm.

As we know, Surface Area ( SA ) = πd 2 , here π = 22/7 = 3.141, d = 2 × 9 cm = 18 cm ∴ SA = 3.141 × (18) 2 = 1017.684 cm 2

Let us find out the surface area of a sphere from circumference.

Finding the surface area of a sphere when the CIRCUMFERENCE is known

Find the surface area of a sphere with a circumference of 36 cm.

Here we will use an alternative formula for the surface area using the circumference. Surface Area ( SA ) = ${\dfrac{C^{2}}{\pi }}$ , here  C = 36 cm, π = 22/7 = 3.141 ∴ SA = (36) 2 ÷ 3.141 = 412.6 cm 2

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How to Find the Surface Area of a Sphere

Last Updated: January 16, 2024 Fact Checked

This article was co-authored by Joseph Meyer . Joseph Meyer is a High School Math Teacher based in Pittsburgh, Pennsylvania. He is an educator at City Charter High School, where he has been teaching for over 7 years. Joseph is also the founder of Sandbox Math, an online learning community dedicated to helping students succeed in Algebra. His site is set apart by its focus on fostering genuine comprehension through step-by-step understanding (instead of just getting the correct final answer), enabling learners to identify and overcome misunderstandings and confidently take on any test they face. He received his MA in Physics from Case Western Reserve University and his BA in Physics from Baldwin Wallace University. There are 8 references cited in this article, which can be found at the bottom of the page. This article has been fact-checked, ensuring the accuracy of any cited facts and confirming the authority of its sources. This article has been viewed 355,952 times.

The surface area of a sphere is the number of square units (cm 2 , square inches, square feet -- whatever your measurement) that are covering the outside of a spherical object. [1] X Research source Discovered by the Greek philosopher and mathematician Aristotle thousands of years ago, the equation is relatively simple, even if its origins are not. To find the surface area of a sphere, use the formula (4πr 2 ), where r = the radius of the circle.

• r, or "radius: The radius is the distance from the center of the sphere to the edge of that sphere.
• π, or "pi:" This incredible number (equalling roughly 3.14) represents the ratio between a circle's circumference and diameter, and is useful in all equations with circles and spheres. It is commonly shortened as π = 3.1416, but there are an infinite number of decimals. [3] X Research source
• 4: For somewhat complex reasons, the surface area of a sphere is always 4 times as large as the area of a circle with the same radius.

• Advanced Tip: If you only know the volume of a sphere, you need to do a little more work to get the radius. Divide the volume by 4π, then multiply that answer by 3. Finally, take the cube root of this answer. [5] X Research source

• If our radius is 5, like above, you would be left with 4 * 25 * π, or 100π.

• 100 * π = 100 * 3.14

• The full answer to the sphere in the pictures is: Surface Area = 314 units 2 .
• The units you use are always the same ones used to measure the radius. If the radius is in meters, the answer will be in meters.
• Advanced Tip: We square the units because area measures how many flat squares we could fit on the surface of the sphere. Say we measure the practice problem in inches. This means on a sphere where r=5, we could fit 314 squares on the surface of the sphere if the sides of every square are 1 inch long.

• 4 * π * 7 2
• Answer: Surface Area = 615.75 centimeters 2 , or 615.75 square centimeters.

• Rotating a circle around its axis (the center point) will produce a sphere. Think of spinning a coin on the table and how it appears to form a sphere. While it won't be explained here, this is where our equation comes from.
• Advanced Tip: Spheres have a smaller surface area per volume than any other shape -- that means it can hold more things in a smaller area than any other shape.

Community Q&A

• If your radius includes a square root, like 3 √ 5, remember to square coefficient squares and the radical. (3 √ 5) 2 becomes 9×5 which gives 45. Thanks Helpful 1 Not Helpful 1

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Expert Interview

Thanks for reading our article! If you’d like to learn more about math, check out our in-depth interview with Joseph Meyer .

• ↑ https://www.mathopenref.com/spherearea.html
• ↑ https://www.omnicalculator.com/math/area-of-sphere
• ↑ http://wonderopolis.org/wonder/what-is-pi/
• ↑ https://www.khanacademy.org/math/basic-geo/basic-geo-area-and-perimeter/area-circumference-circle/a/radius-diameter-circumference
• ↑ http://demonstrations.wolfram.com/RelationOfRadiusSurfaceAreaAndVolumeOfASphere/
• ↑ https://www.cuemath.com/measurement/surface-area-of-sphere/
• ↑ https://www.mathway.com/popular-problems/Basic%20Math/350

About This Article

To find the surface area of a sphere, use the equation 4πr2, where r stands for the radius, which you will multiply by itself to square it. Then, multiply the squared radius by 4. For example, if the radius is 5, it would be 25 times 4, which equals 100. If the problem calls for an exact answer, then leave the answer as 100π. If the answer doesn’t need to be exact, multiply by 3.14 to get the surface area. Be sure to label your answer as the appropriate units squared. If you want to learn how to find the radius of a sphere, keep reading the article! Did this summary help you? Yes No

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Formula Surface Area of a Sphere

Sphere's Surface Area

The surface area of the sphere pictured on the left is:

$$ A = 4\pi \cdot r^2 \\ A = 4\pi \cdot 3^2 \\ A = 36\pi \\ = 113.1 $$

Practice Problems

What is the surface area of a sphere with a diameter of 12 cm?

First, remember that we need to divide the diameter by 2 to get the radius of 6 for our formula:

$$ A = 4\pi \cdot 6^2 \\ A = 4\pi \cdot 6^2 \\ A = 144\pi \\ = 452.4 cm^2 $$

Find the radius of a sphere with a surface area of 196 $$\pi$ square inches?

Substitute the known values into the formula and solve for the radius.

$$ A = 4\pi \cdot r^2 \\ 196\pi = 4\pi \cdot r^2 \\ \frac{196\pi}{4\pi} = \frac{4\pi \cdot r^2}{4\pi} \\ 49 = r^2 \\ r^2 = \sqrt{49} = 7\text{ inches} $$

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Surface Area of a Sphere

In these lessons we will learn

• the formula of the surface area of a sphere.
• how to calculate the surface area of a sphere.
• how to calculate the surface area of a hemisphere.
• how to solve problems about the surface area of spheres.
• how to prove the formula of the surface area of a sphere.

Related Pages Surface Area Of Spheres Worksheet More Geometry Lessons

A sphere is a solid in which all the points on the round surface are equidistant from a fixed point , known as the center of the sphere. The distance from the center to the surface is the radius .

Surface area of a sphere is given by the formula:

Surface Area of sphere = 4πr 2

where r is the radius of the sphere.

Example: Calculate the surface area of a sphere with radius 3.2 cm

Solution: Surface area of sphere = 4π r 2 = 4π (3.2) 2 = 4 × 3.14 × 3.2 × 3.2 = 128.6 cm 2

Worksheet to calculate the surface area of spheres.

How to find the surface area of a sphere?

Example: Find the surface area of a sphere with r = 4 ft

How to calculate the surface area of a sphere given the radius or diameter?

Example: Find the surface area of a sphere with diameter = 22 cm

Surface Area of a hemisphere

A hemisphere is half a sphere, with one flat circular face and one bowl-shaped face.

The surface area of a hemisphere is equal to the area of the curve surface plus the area of the circular base.

Surface area of hemisphere

Example: Calculate the surface area of a hemisphere with radius 4 cm

Solution: Surface area of hemisphere

= 150.86 cm 2

Problems about surface area of spheres

Problem: What is the radius of the sphere given the surface area? Example: What is the radius of a sphere with a surface area of 900π?

Problem: The surface area of a sphere is 5024 square meters. What is the volume of the sphere? Use 3.14 for pi. Round your answer to the nearest cubic meter.

Problem: The radius of a sphere is tripled. a) Describe the effect on the volume. b) Describe the effect on the surface area.

Proof of the formula of the surface area of a sphere

This video demonstrates that the surface area of a sphere equals the area of 4 circles. (It is not a formal proof)

The formula for the surface of a sphere is derived by summing up small ring elements of area along its perimeter. (uses calculus)

These two videos explain the Archimedes method of deriving the surface area of a sphere. Part1:

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Surface Area of Sphere

What is the surface area of a sphere, derivation of surface area of sphere, formula of surface area of sphere, solved examples, practice problems, frequently asked questions.

The surface area of a sphere is the region or area covered by the outer, curved surface of the sphere. 

A sphere is a three-dimensional solid with every point on the surface at equal distances from the center. In simple words, any solid, round object shaped like a ball is a sphere. 

The radius of a sphere is the distance between the surface and the center of the sphere. While one endpoint of radius is on the surface, the other lies at the center of that sphere. 

Read on to know the exact definition of the surface area of a sphere, along with the derivation of the formula. 

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Definition of the Surface Area of a Sphere 

The surface area of a sphere is defined as the region covered by the sphere’s outer surface in three-dimensional space. Since the sphere is curved, its curved surface area is the same as the total area of the sphere. 

It is expressed as:

Surface area $(TSA) =$ Curved Surface Area $(CSA) = 4\pi r^{2}$ square units

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Now that you have understood the surface area of a sphere, it’s time to derive the formula. 

The Greek mathematician Archimedes discovered that the surface area of a sphere is the same as the lateral surface area of a cylinder such that the radius of the sphere is the same as the radius of the cylinder and the height of the cylinder is the same as the diameter of the sphere. According to Archimedes, a sphere can fit into a cylinder so that the height of the cylinder becomes the diameter of the sphere. 

So, if the radius of a sphere and cylinder is “r,” then

Surface Area of Sphere $=$ Lateral Surface Area of Cylinder

We know that the lateral surface area of a cylinder $= 2\pi r h$, 

where “r” is the radius and ‘h’ is the height of the cylinder. 

Now the height of the cylinder = the diameter of the sphere. 

So, $h = 2r$

Substituting the value of “h” as “2r,” the surface area equation of the sphere becomes 

$2\pi r h = 2\pi r(2r) = 4\pi r^{2}$

So, what’s the surface area of a sphere formula?

Surface area of the sphere $= 4\pi r^{2}$, where “r” is the sphere’s radius.

In terms of diameter, when “d” is the diameter of the sphere, the surface area of a sphere is expressed as S $= 4\pi (\frac{d}{2})^{2}$

There are three types of surface area in solids: lateral surface area (LSA), curved surface area (CSA), and total surface area (TSA). 

Curved Surface Area of Sphere

The curved surface area is the sum of areas of all the curved regions of the solid.

The curved surface area of a sphere is the same as the total surface area of the sphere since the sphere is a completely curved shape. Therefore, the formula for the curved surface area of a sphere is 

CSA of a sphere $= 4\pi r^{2}$, where “r” is the sphere’s radius.

Lateral Surface Area of Sphere

The lateral surface area of a solid figure is the area of all the regions except its top and bottom faces.

The lateral surface area of a sphere is expressed as $LSA = 4\pi r^{2}$, where r is the sphere’s radius. 

Total Surface Area of Sphere

The total surface area of a solid includes the area of all sides, including the top and bottom faces or bases.

Since the sphere has no flat surfaces, there is no difference between its curved surface area and the total surface area.

Therefore, the total surface area of a sphere $=$ the curved surface area of a sphere.

Hence, surface area $(TSA) = CSA = 4\pi r^{2}$ square units

How to Find the Surface Area of a Sphere

Let’s understand how to calculate the surface area of a sphere using an example. If the radius of a sphere is 8 cm. What will be its surface area?

The steps to find the surface area of the sphere are given below:

• Step 1: 

Note the radius of the sphere. In the above example, the radius of the sphere is 8 cm. If the diameter is given, divide the diameter by 2 to get the radius.

Now, surface area of sphere $= 4\pi r^{2}$, so after substituting the value of $r = 8$, we get, surface area of sphere $= 4\pi r^{2} = 4 \times 3.14 \times 8^{2} = 4 \times 3.14 \times 64= 803.84$

• Step 3: 

Hence, the surface area of the sphere is $803.84$ $cm^{2}$

1. Calculate the curved surface area of a sphere having a radius of 3 cm. Use $\pi = 3.14$ . 

Solution: 

Curved surface area = Total surface area $= 4 \pi r^{2}$ square units

       $= 4 \times 3.14 \times 3 \times 3$

Therefore, the curved surface area of a sphere$= 113.04$ $cm^{2}$

2. A ball in the shape of a sphere has a surface area of $221.76$ $cm^{2}$ . Calculate its diameter.

Solution:  Let the radius of the sphere be r cm.

We know the surface area of a sphere $=4\pi r^{2}$

        $221.76$ $cm^{2} = 4\pi r^{2}$

                        $r^{2} = 221.76/4 \pi$

  $r = \sqrt{17.64}$

  $r = 4.2$cm

Therefore, the diameter of the sphere $= 4.2 \times 2= ​8.4$ cm.

3. A spherical ball has a surface area of 2464 sq. feet. Find the radius of the ball.

Solution: The surface area of a sphere $= 2464$ sq. feet

We know the surface area of a sphere (SA) $= 4 \pi r^{2}$

       $2464$ $cm^{2} = 4 \pi r^{2}$

                                                      $2464$ $cm^{2} = 4 \times \frac{22}{7} \times r^{2}$

                                                                    $r = 14$ feet

4. Find the ratio between the surface areas of two spheres whose radii are in the ratio of 4:3.

Given that the ratio between the radii of two spheres $= 4:3$

We know that,

The surface area of a sphere $= 4\pi r^{2}$

From the equation, we can say that the surface area of a sphere is directly proportional to the square of its radius.

$\Rightarrow \frac{A_{1}}{A_{2}} = \frac{r^{2}_{1}}{r^{2}_{2}}$

$\Rightarrow \frac{A_{1}}{A_{2}} = \frac{16}{9}$

Therefore, the ratio between the two spheres’ total surface areas is 16:9.

5. Find the cost required to paint a spherical ball with a radius of 10 feet. The painting cost of the ball is $\$4$ per square feet.

Given that the radius of the ball $= 10$ feet

The surface area of a sphere $= 4 \pi r^{2}$ square units

$= 4 \times (3.14) \times (10)^{2}$

$= 1256$ square feet

Hence, the total cost to paint the ball $= 4 \times 1256 = \$5024$

Attend this quiz & Test your knowledge.

If the diameter of a sphere is 16 units, what is the surface area of the sphere?

What's the diameter of a sphere whose surface area is 616 square inches, a solid sphere has a diameter of 28 feet. what is the curved surface area of the sphere, the curved surface area of a sphere is $5544$ $cm^{2}$. what will be the diameter of the sphere, the surface area of a sphere equals lateral surface area of ___ with the same radius and with height same as diameter of sphere..

What is the surface area of a hemisphere?

The surface area of a hemisphere is equal to the sum of its curved surface area and base area. It is expressed as SA(hemisphere) $= 3 \pi r^{2}$ square units.

What is the surface area of a sphere in terms of diameter?

The surface area of a sphere formula in terms of diameter is given as $\pi D^{2}$, where “D” is the sphere’s diameter.

How does the surface area of a sphere change when its radius is halved?

As the surface area of a sphere $= 4\pi r^{2}$ . If we replace “r” with $\frac{r}{2}$, the formula becomes $4\pi (\frac{r}{2})^{2} = \pi r^{2}$, which is one-fourth of the original surface area.

How does the surface area of a sphere change when its radius is tripled?

As the surface area of a sphere $= 4\pi r^{2}$, so, if we replace “r” with $3r$, we get the formula, surface area $= 4\pi (3r)^{2} = 36\pi r^{2}$.

How many surfaces does a sphere have?

A sphere has only one face, which is a curved surface. It does not have any flat faces.

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PROBLEMS ON SURFACE AREA OF SPHERE AND HEMISPHERE

Problem 1 :

If the curved surface area of solid sphere is 98.56 cm 2 , then find the radius of the sphere.

Curved surface area of sphere  =  98.56 c m 2

4 Π r 2   =  98.56

4  ⋅  (22/7)  ⋅  r²  = 98.56

r 2   =  98.56  ⋅  (1/4)  ⋅  (7/22)

r 2   =  98.56  ⋅ ( 1/4)  ⋅  (7/22)

r 2   =  7.84

r  =  √(2.8  ⋅  2.8) 

r  =  2.8 cm

So, radius of the sphere is 2.8 cm.

Problem 2 :

If the curved surface area of the solid hemisphere is 2772 sq.cm, then find its total surface area.

Solution  :

Curved surface area of hemisphere  =  2772 cm 2

2Π r 2   =  2772

2   ⋅ (22/7)  ⋅  r 2   =  2772

r 2   =  2772   ⋅  (1/2 )  ⋅ (7/22)

r 2   =  441

r  =  21

Total surface area of hemisphere   =  3Πr 2

=  3 ⋅ (22/7) ⋅(21) 2

=  4158 cm 2  

Total surface area of sphere = 4158 cm²  

Problem 3 :

Radii of two solid hemispheres are in the ratio 3:5. Find the ratio of their curved surface areas and the ratio of their total surface areas.

Let r₁ and r₂ are the radii of two hemispheres

r 1  : r 2   =  3:5

r 1  / r 2   =  3/5

r 1   =  3r 2 /5

Curved surface area of hemisphere  =   2Πr 2

Ratio of curved surface area of two hemisphere

2 Π r 2  : 2 Π r 2

(3 r 2 /5) 2  : r 2 2

9 : 25

Total surface area of hemisphere =    3Πr²

3Π r 1 2  : 3 Πr 2 2

(3 r₂/5)² : r₂²

Ratio of curved surface area is 9 : 25

Ratio of total surface area is 9 : 25

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A sphere is the collection of points in space which are equidistant from a fixed point. This point is called the center of the sphere. The common distance of the points of the sphere from the center is called the radius .

Spheres are the natural 3-dimensional analog of circles .

Fractions of a sphere

From this formula, we can deduce other sphere-related formulas, such as the volume of a cap cut off by a plane.

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Course: 6th grade   >   Unit 10

• Expressions to find surface area
• Surface area
• Surface area versus volume
• Surface area word problem example

Surface area word problems

• Surface area review
• 3D figures FAQ

• Your answer should be
• an integer, like 6 ‍  
• a simplified proper fraction, like 3 / 5 ‍  
• a simplified improper fraction, like 7 / 4 ‍  
• a mixed number, like 1   3 / 4 ‍  
• an exact decimal, like 0.75 ‍  
• a multiple of pi, like 12   pi ‍   or 2 / 3   pi ‍  

Volume of Sphere Exercises

Volume of sphere practice problems with answers.

You will find ten (10) practice problems below regarding the sphere’s volume. This set of problems should provide you enough practice on how to use the formula to find volume of the sphere.

Use the formula below as a guide. Good luck!

Problem 1: Find the volume of a sphere that has a radius of [latex]5[/latex] yards. Use [latex]\pi=3.14[/latex]. Round your answer to the nearest tenth.

Therefore, the volume of the sphere is approximately [latex]523.3[/latex] [latex]\text{yd}^3[/latex].

Problem 2: The radius of a sphere is [latex]1.5[/latex] centimeters. What is its volume? Use [latex]\pi=3.1416[/latex]. Round your answer to the nearest hundredth.

Therefore, the volume of the sphere is approximately [latex]14.14[/latex] [latex]\text{cm}^3[/latex].

Problem 3: Calculate the volume of a sphere with a radius of [latex]8[/latex] inches. Use [latex]\pi=22/7[/latex]. Round your answer to the nearest whole number.

Therefore, the volume of the sphere is approximately [latex]2,146[/latex] [latex]\text{in}^3[/latex].

Problem 4: Determine the volume of a sphere that has a diameter of [latex]14[/latex] meters. Use [latex]\pi=3.14[/latex]. Round your answer to the nearest hundredth.

Therefore, the volume of the sphere is approximately [latex]1,436.03[/latex] [latex]\text{m}^3[/latex].

Problem 5: If the diameter of a sphere is [latex]7[/latex] feet, what is its volume? Use [latex]\pi=3.1416[/latex]. Round your answer to the nearest thousandth.

Therefore, the volume of the sphere is approximately [latex]179.595[/latex] [latex]\text{ft}^3[/latex].

Problem 6: Find the volume of a sphere whose diameter is [latex]10.5[/latex] centimeters. Use [latex]\pi=22/7[/latex]. Round your answer to the nearest whole number.

Therefore, the volume of the sphere is approximately [latex]606[/latex] [latex]\text{cm}^3[/latex].

Problem 7: The volume of the sphere is [latex]3,719[/latex] cubic kilometers. What is the radius of the sphere? Round your answer to the nearest hundredth. Use [latex]\pi=3.14[/latex].

Therefore, the radius of the sphere is [latex]9.61[/latex] [latex]\text{km}[/latex].

Problem 8: Given that the volume of the sphere is [latex]1,000[/latex] cubic inches. Calculate its radius. Round your answer to the nearest tenth. Use [latex]\pi=22/7[/latex].

Therefore, the radius of the sphere is [latex]6.2[/latex] [latex]\text{in.}[/latex].

Problem 9: The surface area of a sphere is [latex]5,544[/latex] [latex]\text{ft}^2[/latex]. Find the volume of the sphere. Use [latex]\pi=22/7[/latex] as an approximation.

Find the radius of the sphere using the surface area formula.

That means, the radius of the sphere is [latex]21[/latex] [latex]\text{ft}[/latex].

Use the radius from previous step to determine the volume of the sphere.

Therefore, the volume of the sphere is [latex]38,808[/latex] [latex]\text{ft}^3[/latex].

Problem 10: If the surface area of a sphere is [latex]9,856[/latex] [latex]\text{m}^2[/latex], what is its volume? Round you answer to one decimal place. Use [latex]\pi=22/7[/latex] as an approximation.

That means the radius of the sphere is [latex]28[/latex] [latex]\text{m}[/latex].

Now use the value of the radius to calculate the volume of the sphere.

Therefore, the volume of the sphere is [latex]91,989.3[/latex] [latex]\text{m}^3[/latex].

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• Volume of Sphere Formula