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Hypotenuse Calculator

Calculate the length of the hypotenuse in a right triangle using two sides or one side and one angle with the calculator below.

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How to find the hypotenuse of a right triangle, formula one: given two legs, formula two: given one leg and the adjacent angle, formula three: given one leg and the opposite angle, formula four: given one leg and the area, how to find the hypotenuse for a 45 45 90 right triangle, how to find the hypotenuse for a 30 60 90 right triangle.

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The hypotenuse is the longest side of a right triangle and is always opposite the 90° right angle.

The illustration below is a right triangle . The hypotenuse is labeled as c , and the opposite 90° angle is typically symbolized with the square drawn at the vertex.

The two shorter side lengths (often called the legs of a right triangle), are labeled as a & b , and are opposite the interior angles labeled α & β .

You can solve the hypotenuse using a little trigonometry and one of the following formulas.

The Pythagorean theorem can be used to find the hypotenuse using the length of the two legs. The Pythagorean theorem states a² + b² = c² .

This formula can be rewritten to solve for the hypotenuse c :

c = a² + b²

Thus, the hypotenuse c is equal to the square root of the quantity leg a squared plus leg b squared.

Given the length of one leg and the adjacent angle, you can find the hypotenuse using the formula:

c = a / cos(β)

The hypotenuse c is equal to side a divided by the cosine of the adjacent angle β .

Given the length of one leg and the opposite angle, you can find the hypotenuse using the Law of Sines with the formula:

c = a / sin(α)

The hypotenuse c is equal to side a divided by the sine of the opposite angle α .

If you know the length of one of the legs and the triangle area , you can find the length of the hypotenuse by using the area formula to solve for the length of the other leg, then using the Pythagorean theorem.

Given the formula to find the area of a right triangle, start by finding the length of the other leg:

A = 1 / 2 ab

You can rearrange this formula to solve for the length of leg b like this:

The length of leg b is equal to 2 times the area A divided by the length of leg a .

Then, you can use the length of legs a and b to find the hypotenuse using the first formula above:

A 45 45 90 triangle is a special right triangle that is also an isosceles triangle. As the name implies, a 45 45 90 has two 45° interior angles and one right interior angle.

Since this is an isosceles triangle , both legs are equal in length, so you can find the length of the hypotenuse of a 45 45 90 right triangle using a simplified formula derived from the Pythagorean theorem.

c = a√2

The length of hypotenuse c is equal to the length of leg a times the square root of 2.

A 30 60 90 triangle is a special right triangle that has one 30° interior angle, one 60° interior angle, and one right interior angle.

In this special case, the length of the hypotenuse is always equal to two times the length of the shortest leg a of the triangle. Note, the shortest leg will always be opposite the 30° interior angle.

So, given the length of the shortest leg a , the formula to find the hypotenuse is:

The length of hypotenuse c is equal to 2 times the length of leg a .

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Right Triangle Calculator

Please provide values for any three of the six fields below. At least one of those values must be a side length.

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About the Right Triangle Calculator

This right triangle calculator lets you calculate the length of the hypotenuse or a leg or the area of a right triangle. For each case, you may choose from different combinations of values to input.

Also, the calculator will give you not just the answer, but also a step by step solution. So you can use it as a great tool to learn about right triangles.

Usage Guide

I. valid inputs.

The calculator needs exactly two inputs, at least one of which must be a side (a leg or the hypotenuse).

All inputs can be in any of the three number formats listed below.

Note — The input for each of the two angles must be between 0 \hspace{0.2em} 0 \hspace{0.2em} 0 and 90 \hspace{0.2em} 90 \hspace{0.2em} 90 .

ii. Example

If you would like to see an example of the calculator's working, just click the "example" button.

iii. Solutions

As mentioned earlier, the calculator won't just tell you the answer but also the steps you can follow to do the calculation yourself. The "show/hide solution" button would be available to you after the calculator has processed your input.

We would love to see you share our calculators with your family, friends, or anyone else who might find it useful.

By checking the "include calculation" checkbox, you can share your calculation as well.

Here's a quick overview of what a right triangle is a few of the basic concepts related to it.

Right Triangles

A right triangle (or right-angled triangle) is a triangle in which one of the three internal angles is a right angle ( 90 ° \hspace{0.2em} 90 \degree \hspace{0.2em} 90° ).

The longest side in a right triangle (known as the hypotenuse) is the side opposite the right angle.

Pythagorean Theorem

Now one feature of right triangles that makes them so useful and important is that they obey the pythagorean theorem.

And according to the Pythagorean theorem , the square of the hypotenuse is equal to the sum of the squares of the other two sides (called legs). So, for the triangle above –

The two legs of a right triangle measure 5  cm \hspace{0.2em} 5 \text{ cm} \hspace{0.2em} 5  cm and 12  cm \hspace{0.2em} 12 \text{ cm} \hspace{0.2em} 12  cm in length. Find the length of the hypotenuse.

If x \hspace{0.2em} x \hspace{0.2em} x denotes the length of the hypotenuse, according the pythagorean theorem —

Taking the square root on both sides, we have

So the hypotenuse has a length of 13  cm \hspace{0.2em} 13 \text{ cm} \hspace{0.2em} 13  cm .

Trigonometric Ratios

Another concept that makes right triangles great for the study of triangles is that of trigonometric ratios. Here's a brief explanation.

We'll start with the figure below.

Now, when we talk about trigonometric ratios, those ratios are with respect to a reference angle. And that reference angle can be any of the two acute angles in a right triangle.

Also, as the figure shows, we have special names for the two legs. The one opposite to the angle is termed "opposite" and the one adjacent to it is called "adjacent".

Trigonometric ratios are ratios between the side lengths of a right triangle. And the value of a trigonometric ratio depends on the reference angle alone.

Here's a table listing the six trigonometric ratios.

As mentioned earlier, values for trigonometric ratios depend only on the reference angle ( θ ) \hspace{0.2em} (\theta) \hspace{0.2em} ( θ ) . This is crucial, as you'll see in the second example below.

In △ A B C \hspace{0.2em} \triangle ABC \hspace{0.2em} △ A BC , ∠ C = 90 ° \hspace{0.2em} \angle C = 90 \degree \hspace{0.2em} ∠ C = 90° , ∠ B = 40 ° \hspace{0.2em} \angle B = 40 \degree \hspace{0.2em} ∠ B = 40° , and A C = 5  in \hspace{0.2em} AC = 5 \text{ in} \hspace{0.2em} A C = 5  in . Find the lengths of A B \hspace{0.2em} AB \hspace{0.2em} A B and B C \hspace{0.2em} BC \hspace{0.2em} BC .

Let's start with a rough labeled sketch of the triangle.

Next, we know

So, for the given triangle

Now, because the value of a trigonometric ratio depends only on the angle, sin ⁡ 40 ° \hspace{0.2em} \sin 40 \degree \hspace{0.2em} sin 40° would be a constant. As a calculator would tell you, sin ⁡ 40 ° ≈ 0.59 \hspace{0.2em} \sin 40 \degree \hspace{0.1em} \approx \hspace{0.25em} 0.59 \hspace{0.2em} sin 40° ≈ 0.59 .

Substituting the value of sin ⁡ 40 ° \hspace{0.2em} \sin 40 \degree \hspace{0.2em} sin 40° into the equation above, we have

Substituting tan ⁡ 40 ° ≈ 0.73 \hspace{0.2em} \tan 40 \degree \hspace{0.1em} \approx \hspace{0.25em} 0.73 \hspace{0.2em} tan 40° ≈ 0.73 ,

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Course: high school geometry   >   unit 5, hypotenuse, opposite, and adjacent.

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Hypotenuse Calculator

What is a hypotenuse of a triangle, the hypotenuse of a triangle formula, using a hypotenuse calculator: finding the hypotenuse of a right triangle.

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To use the right triangle calculator, Select the unknown values, Enter the known value, and Click Calculate. 

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• What is a right triangle? 

How to find the sides or angles of the right triangle?

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Right Triangle Calculator 

Right triangle calculator is an all-at-one place tool that finds almost every basic unknown quantity of a right triangle. You can find any angle or side using this tool.

What is a right triangle? 

A triangle that has one angle at 90 degrees is called a right-angled triangle. The side that is opposite the right angle ( 90 degrees) is called hypotenuse and the other two are the adjacent and the opposite sides. 

The three sides of a right triangle are usually referred to by variables commonly c for hypotenuse and a and b for the other sides. 

One of the properties of a right triangle is the Pythagorean theorem , which states that in any right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides. This can be written as:

a² + b² = c²

• a and b are the lengths of the two sides that form the right angle, and
• c is the length of the hypotenuse.

Given different pieces of information, there are various methods to find the sides and angles of a right triangle.

To find the sides from sides:

The main method to find the sides of a triangle is using the Pythagorean theorem. Input the known sides and change the formula to separate the unknown value on one side. 

Example: 

A right triangle has the following data

Side b = 4cm

Side a = 3cm 

Find the side c.

Since side c is the hypotenuse, no change in the Pythagorean theorem is required. Just enter the values and solve. 

(3)² + (4)² = c²

9 + 16 = c²

25 = c²

The length of the hypotenuse is 5

To find the angles from angles:

The total sum of the angles of a triangle is always 180 degrees. Since a right triangle always has a 90 -degree angle, the sum of the other two angles is also 90. I.e. 

Hypotenuse (90) + Other two angles (90) = Total angle (180) 

A right triangle has an angle A of 65 degrees. Find angle B. 

We know the measurement of the right angle i.e. 90 degrees. 

So to find angle B,

90 + 65 + B = 180

155 + B = 180 

B = 180 - 155

Or you can simply subtract the value of angle A or B whichever is provided from 90 to find the unknown angle.

To find a side and an angle:

There are many possibilities for this case. However, all of the questions can be solved using trigonometric functions. In trigonometry 

     a / sin(a) = b / sin(b) = c / sin(c)

• Sin (a) = a / c
• Cos (a) = b / c
• Tan (a) = a / b
• Cot (a) = b / a
• Sin (b) = b / c
• Cos (b) = a / c
• Tan (b) = b / a
• Cot (b) = a / b

These rules can be used to find any length or angle if provided any two quantities. 

According to the given data, look for the right formula. Say that you have two angles (A, B) and a side (a) and you have to find the side(b). Use the formula 

a/ sin (a) = b / sin (b)

b  = (a)(sin (b)) / sin (a)

This formula can also be used when two sides and one angle are provided to find the second angle.

A Right triangle has an angle of 30 degrees and an adjacent side ( b ) of length 5. Find the opposite side ( a ) and the hypotenuse ( c ).

• To find a (the opposite side), Use the tangent function, tan(θ) = a/b . So a = btan(30) = 5tan(30) = 2.5 .
• To find c (the hypotenuse), Use the cosine function, cos(θ) = b/c . So c = b/cos(30) = 5/cos(30) = approximately 5.77 .
• The other acute angle in the triangle would be 90 - 30 = 60 degrees (since the sum of the two acute angles in a right triangle is 90 degrees).

A Right triangle has a hypotenuse ( c ) of length 10 and an adjacent side ( b ) of length 8 . Find the opposite side ( a ) and the angle ( θ ).

• To find a (the opposite side), Use the Pythagorean theorem: a² = c² - b² = 10² - 8² = 100 - 64 = 36 . Taking the square root of both sides gives a = √36 = 6 .
• To find θ (the angle), Use the cosine function, cos(θ) = b/c . So θ = arccos(b/c) = arccos(8/10) = arccos(0.8) = approximately 36.87 degrees.
• The other acute angle in the triangle would be 90 - 36.87 = 53.13 degrees.
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Right Triangles

Rules, Formula and more

Pythagorean Theorem

The sum of the squares of the lengths of the legs equals the square of the length of the hypotenuse .

Usually, this theorem is expressed as $$ A^2 + B^2 = C^2 $$ .

Right Triangle Properties

A right triangle has one $$ 90^{\circ} $$ angle ($$ \angle $$ B in the picture on the left) and a variety of often-studied formulas such as:

• The Pythagorean Theorem
• Trigonometry Ratios (SOHCAHTOA)
• Pythagorean Theorem vs Sohcahtoa (which to use)

SOHCAHTOA only applies to right triangles ( more here ) .

A Right Triangle's Hypotenuse

The hypotenuse is the largest side in a right triangle and is always opposite the right angle.

In the triangle above, the hypotenuse is the side AB which is opposite the right angle, $$ \angle C $$.

Online tool calculates the hypotenuse (or a leg) using the Pythagorean theorem.

Practice Problems

Below are several practice problems involving the Pythagorean theorem, you can also get more detailed lesson on how to use the Pythagorean theorem here .

Find the length of side t in the triangle on the left.

Substitute the two known sides into the Pythagorean theorem's formula : A² + B² = C²

What is the value of x in the picture on the left?

Set up the Pythagorean Theorem : 14 2 + 48 2 = x 2 2,500 = X 2

$$ x = \sqrt{2500} = 50 $$

$$ x^2 = 21^2 + 72^2 \\ x^2= 5625 \\ x = \sqrt{5625} \\ x =75 $$

Find the length of side X in the triangle on on the left?

Substitue the two known sides into the pythagorean theorem's formula : $$ A^2 + B^2 = C^2 \\ 8^2 + 6^2 = x^2 \\ x = \sqrt{100}=10 $$

What is x in the triangle on the left?

x 2 + 4 2 = 5 2 x 2 + 16 = 25 x 2 = 25 - 16 = 9 x = 3

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How to Find the Length of the Hypotenuse

Last Updated: April 13, 2024 Fact Checked

This article was co-authored by Grace Imson, MA . Grace Imson is a math teacher with over 40 years of teaching experience. Grace is currently a math instructor at the City College of San Francisco and was previously in the Math Department at Saint Louis University. She has taught math at the elementary, middle, high school, and college levels. She has an MA in Education, specializing in Administration and Supervision from Saint Louis University. There are 11 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 1,514,355 times.

All right triangles have one right (90-degree) angle, and the hypotenuse is the side that is opposite or the right angle, or the longest side of the right triangle. [1] X Research source The hypotenuse is the longest side of the triangle, and it’s also very easy to find using a couple of different methods. This article will teach you how to find the length of the hypotenuse using the Pythagorean theorem when you know the length of the other two sides of the triangle. It will then teach you to recognize the hypotenuse of some special right triangles that often appear on tests. It will finally teach you to find the length of the hypotenuse using the Law of Sines when you only know the length of one side and the measure of one additional angle.

Using the Pythagorean Theorem

• Right angles are often notated in textbooks and on tests with a small square in the corner of the angle. This special mark means "90 degrees."

• If your triangle has sides of 3 and 4, and you have assigned letters to those sides such that a = 3 and b = 4, then you should write your equation out as: 3 2 + 4 2 = c 2 .

• If a = 3, a 2 = 3 x 3, or 9. If b = 4, then b 2 = 4 x 4, or 16.
• When you plug those values into your equation, it should now look like this: 9 + 16 = c 2 .

• In our example, 9 + 16 = 25 , so you should write down 25 = c 2 .

• In our example, c 2 = 25 . The square root of 25 is 5 ( 5 x 5 = 25 , so Sqrt(25) = 5 ). That means c = 5 , the length of our hypotenuse!

Joseph Meyer

Use this visual trick to understand the Pythagorean Theorem. Imagine a right triangle with squares constructed on each leg and the hypotenuse. by rearranging the smaller squares within the larger square, the areas of the smaller squares (a² and b²) will add up visually to the area of the larger square (c²).

Finding the Hypotenuse of Special Right Triangles

• The first Pythagorean triple is 3-4-5 (3 2 + 4 2 = 5 2 , 9 + 16 = 25). When you see a right triangle with legs of length 3 and 4, you can instantly be certain that the hypotenuse will be 5 without having to do any calculations.
• The ratio of a Pythagorean triple holds true even when the sides are multiplied by another number. For example a right triangle with legs of length 6 and 8 will have a hypotenuse of 10 (6 2 + 8 2 = 10 2 , 36 + 64 = 100). The same holds true for 9-12-15 , and even 1.5-2-2.5 . Try the math and see for yourself!
• The second Pythagorean triple that commonly appears on tests is 5-12-13 (5 2 + 12 2 = 13 2 , 25 + 144 = 169). Also be on the lookout for multiples like 10-24-26 and 2.5-6-6.5 .

• To calculate the hypotenuse of this triangle based on the length of one of the legs, simply multiply the leg length by Sqrt(2).
• Knowing this ratio comes in especially handy when your test or homework question gives you the side lengths in terms of variables instead of integers.

• If you are given the length of the shortest leg (opposite the 30-degree angle,) simply multiply the leg length by 2 to find the length of the hypotenuse. For instance, if the length of the shortest leg is 4 , you know that the hypotenuse length must be 8 .
• If you are given the length of the longer leg (opposite the 60-degree angle,) multiply that length by 2/Sqrt(3) to find the length of the hypotenuse. For instance, if the length of the longer leg is 4 , you know that the hypotenuse length must be 4.62 .

Finding the Hypotenuse Using the Law of Sines

• To find the sine of an 80 degree angle, you will either need to key in sin 80 followed by the equal sign or enter key, or 80 sin . (The answer is -0.9939.)
• You can also type in "sine calculator" into a web search, and find a number of easy-to-use calculators that will remove any guesswork. [10] X Research source

• The Law of Sines can actually be used to solve any triangle, but only a right triangle will have a hypotenuse.

• For example, if you know that A = 40 degrees , then B = 180 – (90 + 40) . Simplify this to B = 180 – 130 , and you can quickly determine that B = 50 degrees .

• To continue our example, let's say that the length of side a = 10. Angle C = 90 degrees, angle A = 40 degrees, and angle B = 50 degrees.

• Using our example, we find that sin 40 = 0.64278761. To find the value of c, we simply divide the length of a by this number, and learn that 10 / 0.64278761 = 15.6 , the length of our hypotenuse!

Finding the Hypotenuse Using the Area

Practice Problems and Answers

Expert Q&A

You Might Also Like

• ↑ http://www.mathsisfun.com/definitions/hypotenuse.html
• ↑ https://www.mathsisfun.com/definitions/pythagoras-theorem.html
• ↑ https://www.mathsisfun.com/pythagoras.html
• ↑ https://www.learnalberta.ca/content/memg/division03/pythagorean%20theorem/index.html
• ↑ https://www.mathsisfun.com/pythagorean_triples.html
• ↑ https://www.ck12.org/book/ck-12-precalculus-concepts/section/4.3/
• ↑ https://www.mathsisfun.com/definitions/sine.html
• ↑ http://www.rapidtables.com/calc/math/Sin_Calculator.htm
• ↑ http://www.mathsisfun.com/algebra/trig-sine-law.html
• ↑ https://www.cut-the-knot.org/pythagoras/cosine2.shtml
• ↑ https://www.mathsisfun.com/algebra/trig-sine-law.html

About This Article

If you need to find the length of the hypotenuse of a right triangle, you can use the Pythagorean theorem if you know the length of the other two sides. Square the length of the 2 sides, called a and b, then add them together. Take the square root of the result to get the hypotenuse. If you want to learn how to find the hypotenuse using trigonometric functions, keep reading the article! Did this summary help you? Yes No

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Table of Contents

Last modified on August 3rd, 2023

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Hypotenuse of a triangle, what is the hypotenuse of a triangle.

A hypotenuse is the longest side of a right triangle. It is the side opposite the right angle (90°). The word ‘hypotenuse’ came from the Greek word ‘hypoteinousa’, meaning ‘stretching under’, where ‘hypo’ means ‘under’, and ‘teinein’ means ‘to stretch’.

How to Find the Hypotenuse of a Right Triangle

a) When Base and Height are Given

To calculate the hypotenuse of a right or right-angled triangle when its corresponding base and height are known, we use the given formula.

By Pythagorean Theorem,

(Hypotenuse) 2 = (Base) 2 + (Height) 2

Hypotenuse = √(Base) 2 + (Height) 2

Thus, mathematically, hypotenuse is the sum of the square of base and height of a right triangle.

The above formula is also written as,

c = √a 2 + b 2 , here c = hypotenuse, a = height, b = base

Let us solve some problems to understand the concept better.

Problem: Finding the hypotenuse of a right triangle, when the BASE and the HEIGHT are known.

What is the length of the hypotenuse of a right triangle with base 8m and height 6m.

As we know, c = √a 2 + b 2 , here a = 6m, b = 8m = √(6) 2 + (8) 2 = √36 + 64 = √100 = 10m

b) When Length of a Side and its Opposite Angle are Given

To find the hypotenuse of a right triangle when the length of a side and its opposite angle are known, we use the given formula, which is called the Law of sines.

c = a/sin α = b/sin β, here c = hypotenuse, a = height, b = base, α = angle formed between hypotenuse and base, β = angle formed between hypotenuse and height

Problem: Finding the hypotenuse of a right triangle, when the LENGTH OF A SIDE and its OPPOSITE ANGLE is known.

Here, we will use the Law of sines formula, c = a/sin α, here a = 12, α = 30° = 12/ sin 30° = 12 x 2 = 24 units

Using the Law of sines formula, c = b/sin β, b = 4, β = 60° = 4/ sin 60° = 8/√3 units

c) When the Area and Either Height or Base are Known

To determine the hypotenuse of a right triangle when the height or base is known, we use the Pythagorean Theorem to derive the formula as shown below:

As we know from the Pythagorean Theorem

c = √(a) 2 + (b) 2 …..(1), here c = hypotenuse, a = height, b = base

Area of right triangle (A) = a x b/2

 b = area x 2/a …… (2)

a = area x 2/b …… (3)

Putting (2) in (1) we get,

c = √(a 2 + (area x 2/a) 2 )

Putting (3) in (1) we get,

c = √(b 2 + (area x 2/b) 2 )

Problem: Finding the hypotenuse of a right triangle, when the AREA and one SIDE are known.

What is the length of the hypotenuse of a right triangle with area 20m 2 and height 6m.

As we know, c = √(a 2 + (area x 2/a) 2 ), here area = 20m 2 , a = 6m  = √6 2 + (20 x 2/6) 2 )  =√80.35 = 8.96 m

What is the length of the hypotenuse of a right triangle with area 14cm 2 and base 9cm.

As we know, c = √(b 2 + (area x 2/b) 2 ), here area = 14cm 2 , b = 9cm = √9 2 + (14 x 2/9) 2 ) = √45.67 = 6.75 m

How to Find the Hypotenuse of a Right Isosceles Triangle

To derive the formula for finding the hypotenuse of a right isosceles triangle we use the Pythagorean Theorem.

As we know,

c = √a 2 + b 2

Let the length of the two equal sides be x, such that (a = b = x)

c =√x 2 + x 2

What is the length of the hypotenuse of a right isosceles triangle with two equal sides measuring 5.5 cm each.

As we know, c = √2x, here x = 5.5 = √2 x 5.5 = 7.77 cm

Find the measure of the length of the hypotenuse of a 45-45-90 triangle with one of the two equal sides measuring 9 cm.

As a 45-45-90 triangle is a right isosceles triangle, we can apply the formula of right isosceles triangle for calculation of area As we know, c = √2x, here x = 9 cm =√2 x 9 = 12.72 cm

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Pythagoras' Theorem

Over 2000 years ago there was an amazing discovery about triangles:

When a triangle has a right angle (90°) ...

... and squares are made on each of the three sides, ...

... then the biggest square has the exact same area as the other two squares put together!

It is called "Pythagoras' Theorem" and can be written in one short equation:

a 2 + b 2 = c 2

• c is the longest side of the triangle
• a and b are the other two sides

The longest side of the triangle is called the "hypotenuse", so the formal definition is:

In a right angled triangle: the square of the hypotenuse is equal to the sum of the squares of the other two sides.

Let's see if it really works using an example.

Example: A "3, 4, 5" triangle has a right angle in it.

Why Is This Useful?

If we know the lengths of two sides of a right angled triangle, we can find the length of the third side . (But remember it only works on right angled triangles!)

How Do I Use it?

Write it down as an equation:

Then we use algebra to find any missing value, as in these examples:

Example: Solve this triangle

Read Builder's Mathematics to see practical uses for this.

Also read about Squares and Square Roots to find out why √ 169 = 13

Example: Solve this triangle.

Example: what is the diagonal distance across a square of size 1.

It works the other way around, too: when the three sides of a triangle make a 2 + b 2 = c 2 , then the triangle is right angled.

Example: Does this triangle have a Right Angle?

Does a 2 + b 2 = c 2 ?

• a 2 + b 2 = 10 2 + 24 2 = 100 + 576 = 676
• c 2 = 26 2 = 676

They are equal, so ...

Yes, it does have a Right Angle!

Example: Does an 8, 15, 16 triangle have a Right Angle?

Does 8 2 + 15 2 = 16 2 ?

• 8 2 + 15 2 = 64 + 225 = 289 ,
• but 16 2 = 256

So, NO, it does not have a Right Angle

Yes, it does!

So this is a right-angled triangle

And You Can Prove The Theorem Yourself !

Get paper pen and scissors, then using the following animation as a guide:

• Draw a right angled triangle on the paper, leaving plenty of space.
• Draw a square along the hypotenuse (the longest side)
• Draw the same sized square on the other side of the hypotenuse
• Draw lines as shown on the animation, like this:

• Cut out the shapes
• Arrange them so that you can prove that the big square has the same area as the two squares on the other sides

Another, Amazingly Simple, Proof

Here is one of the oldest proofs that the square on the long side has the same area as the other squares.

Watch the animation, and pay attention when the triangles start sliding around.

You may want to watch the animation a few times to understand what is happening.

The purple triangle is the important one.

We also have a proof by adding up the areas .

The hypotenuse is the largest side of a right triangle. It is a side opposite to the right angle in a right triangle. The Pythagoras theorem defines the relationship between the hypotenuse and the other two sides of the right triangle, the base, and the perpendicular side. The square of the hypotenuse is equal to the sum of the squares of the base and the perpendicular side of the right triangle.

The Pythagoras theorem has given the Pythagorean triplets and the largest value in Pythagorean triplets is the hypotenuse. Let us learn more about the hypotenuse in this article.

What is a Hypotenuse?

The hypotenuse is the longest side of a right-angled triangle . It is represented by the side opposite to the right angle. It is related to the other sides of the right triangle by the Pythagoras theorem . The square of the measure of the hypotenuse is equal to the sum of the squares of the other two sides of the right triangle. The hypotenuse can be easily recognized in a right triangle as the largest side.

Hypotenuse Definition: In a right-angled triangle, the longest side or the side opposite to the right angle is termed hypotenuse . The hypotenuse is related to the base and the altitude of the triangle , by the formula: Hypotenuse 2 = Base 2 + Altitude 2 . Let us look at the below real-world examples of a hypotenuse in right triangle-shaped objects.

Hypotenuse Equation

To derive an equation or a formula of the hypotenuse , years ago there was an interesting fact revealed about triangles. Hypotenuse equation : The fact states that with a right-angled triangle or a triangle with a 90º angle, squares can be framed using each of the three sides of the triangle. After putting squares against each side, it was observed that the biggest square has the exact same area as the other two squares. To simplify the whole observation, it was later put in a short equation that can also be called a hypotenuse equation.

So, the hypotenuse equation = a 2 + b 2 = c 2 , where c is the length of the hypotenuse and a and b are the other two sides of the right-angled triangle.

Now, look at the image given below to understand the derivation of the above formula. Here we have a = Perpendicular, b = Base, c = Hypotenuse.

Tips and Tricks on Hypotenuse:

The following points will help you to get a better understanding of the hypotenuse and its relation to the other two sides of the right triangle.

• The Pythagoras theorem states that in a right-angled triangle, the square of the hypotenuse (longest side) is equal to the sum of the squares of the other two sides (base and perpendicular ).
• This is represented as: Hypotenuse 2 = Base 2 + Perpendicular 2 .
• Hypotenuse equation is a 2 + b 2 = c 2 . Here, a and b are the legs of the right triangle and c is the hypotenuse.
• The hypotenuse leg theorem states that two triangles are congruent if the hypotenuse and one leg of one right triangle are congruent/equal to the other right triangle's hypotenuse and leg side.

How to Find Hypotenuse?

To find the length of the hypotenuse of a triangle, we will be using the above equation. For that, we should know the values of the base and perpendicular of the triangle. For example, in a right triangle, if the length of the base is 3 units, and the length of the perpendicular side is 4 units, then the length of the hypotenuse can be found by using the formula Hypotenuse 2 = Base 2 + Perpendicular 2 . By substituting the values of the base and perpendicular, we get, Hypotenuse 2 = 3 2 + 4 2 = 9 + 16 = 25. This implies that the length of the hypotenuse is 5 units. This is how we can easily find the length of the hypotenuse by using the hypotenuse equation.

Follow the steps given below to find the hypotenuse length in a right-angled triangle:

• Step 1: Identify the values of base and perpendicular sides.
• Step 2: Substitute the values of base and perpendicular in the formula: Hypotenuse 2 = Base 2 + Perpendicular 2 .
• Step 3: Solve the equation and get the answer.

Let us consider one more example to find the hypotenuse of a triangle. The longest side of the triangle is the hypotenuse and the other two sides of the right triangle are the perpendicular side with a measure of 8 inches, and the base with a measure of 6 inches.

The following formula is helpful to calculate the measure of the hypotenuse → (Hypotenuse) 2 = (Base) 2 + (Perpendicular) 2 = 6 2 + 8 2 = 36 + 64 = 100. This implies, Hypotenuse = √100 = 10 inches. Also, any of the other two sides, the base or the perpendicular side can be easily calculated for the given value of the hypotenuse using the same equation.

Hypotenuse Theorem

The hypotenuse can be related to the other two sides of the right-angled triangle by the Pythagoras theorem. The Pythagoras theorem states that the square of the hypotenuse is equal to the sum of the squares of the base of the triangle, and the square of the altitude of the triangle. Among the three sides of the right triangle, the hypotenuse is the largest side, and Hypotenuse 2 = Base 2 + Altitude 2 . This is known as the hypotenuse theorem . The lengths of the hypotenuse, altitude, and base of the triangle, are together defined as a set called the Pythagorean triplets. A few examples of Pythagorean triples are (5, 4, 3), (10, 8, 6), and (25, 24, 7).

Challenging Questions:

Having understood the concepts related to the hypotenuse of a triangle, now try out these two challenging questions.

• A 5 meters ladder stands on horizontal ground and reaches 3 m up a vertical wall. How far is the foot of the ladder from the wall?
• Town B is 9 km north and 16 km west of town A. What is the shortest distance to go from town A to town B?

► Related Topics:

Check these articles related to the concept of the hypotenuse of a triangle.

• Hypotenuse Calculator
• Area of Right Triangle
• Properties of Triangle

Hypotenuse Examples

Example 1: Find the value of the longest side of a bread slice that is in the shape of a right-angle triangle with a given perpendicular height of 12 inches and the base of 5 inches.

Given dimensions are perpendicular (P) = 12 inches, and base (B) = 5 inches. Putting the given dimensions in the formula H 2 = B 2 + P 2 , we get, H 2 = 5 2 + 12 2 H = √{25+144} = √169 inches H = 13 inches. Therefore the length of the hypotenuse (longest side) of the bread slice is 13 inches.

Example 2: In a right triangle, the hypotenuse is 5 units, and the perpendicular is 4 units. Find the measure of the base of the triangle.

Given dimensions are perpendicular (P) = 4 units, and hypotenuse (H) = 5 units. We know that (H) 2 = (B) 2 + (P) 2 ⇒ (B) 2 = (H) 2 - (P) 2 . Putting the given dimensions in the formula, we get, B 2 = (5) 2 - (4) 2 B = √{25-16} B = √9 = 3 units Therefore, the length of the base is 3 units.

Example 3: How to find the missing hypotenuse of a triangle with base = 7 units and perpendicular = 24 units?

Given dimensions are base (B) = 7 units and perpendicular (P) = 24 units. To find the hypotenuse (H), we will use the equation: (H) 2 = (B) 2 + (P) 2 . Putting the given dimensions in the equation, we get, H 2 = (7) 2 + (24) 2 B = √{49+576} B = √625 = 25 units Therefore, the length of the hypotenuse is 25 units.

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Practice Questions on Hypotenuse

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FAQs on Hypotenuse

What is the meaning of hypotenuse.

In mathematics, the hypotenuse of a triangle is defined as the longest side of a right triangle. It is the side opposite to the 90-degree angle . It is equal to the square root of the sum of the squares of the other two sides.

What is the Length of the Hypotenuse?

The length of the hypotenuse is greater than the lengths of the other two sides of a right triangle. The square of the hypotenuse length is equal to the sum of squares of the other two sides of the triangle. Mathematically, it can be expressed in the form of an equation as Hypotenuse 2 = Base 2 + Perpendicular 2 .

What is the Hypotenuse Leg Theorem?

The hypotenuse leg theorem states that two right triangles are congruent if the lengths of the hypotenuse and any one of the legs of a triangle are equal to the hypotenuse and the leg of the other triangle.

How to Find the Missing Hypotenuse?

The missing hypotenuse can be easily known if we know the lengths of the other two sides by using the hypotenuse equation: Hypotenuse 2 = Base 2 + Perpendicular 2 . For example, if the base and perpendicular of a right triangle measure 6 units and 8 units respectively, then the hypotenuse is equal to:

Hypotenuse 2 = 6 2 + 8 2

Therefore, hypotenuse = 10 units.

How do you Find the Hypotenuse of a Triangle?

By using the Pythagorean theorem (Hypotenuse) 2 = (Base) 2 + (Altitude) 2 , we can calculate the hypotenuse. If the values of the other two sides are known, the hypotenuse can be easily calculated with this formula.

How do you Find the Longest Side of a Triangle?

The hypotenuse is termed as the longest side of a right-angled triangle. To find the longest side we use the hypotenuse theorem, (Hypotenuse) 2 = (Base) 2 + (Altitude) 2 . For example, a bread slice is given in the shape of a right-angled triangle. If the base is 4 inches and the height is 3 inches, then the hypotenuse is (H) 2 = (4) 2 + (3) 2 = √{16+9} = √25 = 5 inches.

How to Find Hypotenuse with Angle and Side?

If an angle and a side are known, then we can calculate hypotenuse by applying the formula of trigonometric ratios . If A is the angle known, then we have,

• sin A = Perpendicular/Hypotenuse
• cos A = Base/Hypotenuse

So, if the length of the base is given, then the cos formula can be used and if height is known then the sin formula can be used to find the hypotenuse length.

What is the Difference between the Hypotenuse and Other Sides of a Triangle?

The hypotenuse is the largest side of the triangle. The other two sides are the base and the altitude of the right triangle. These are related to each other with the formula (Hypotenuse) 2 = (Base) 2 + (Altitude) 2 .

How is the Hypotenuse Related to the Right Angle?

The hypotenuse is the side opposite to the right angle. The hypotenuse is the largest side of a right triangle and is drawn opposite to the largest angle, which is the right angle.

Can a Hypotenuse be Drawn for Any Triangle?

The hypotenuse can be drawn only for a right triangle, and not for any other triangle. The side opposite to the 90° angle is the hypotenuse. And since a right angle is there in a right triangle, it has a hypotenuse.

How to Calculate Hypotenuse?

The formula to calculate the hypotenuse is (Hypotenuse) 2 = (Base) 2 + (Altitude) 2 . The largest side of the right triangle is the hypotenuse, and it can be calculated if the other two sides are known.

Right Triangle Trigonometry Calculator

Basics of trigonometry, right triangles trigonometry calculations, example of right triangle trigonometry calculations with steps, more trigonometry and right triangles calculators (and not only).

The right triangle trigonometry calculator can help you with problems where angles and triangles meet: keep reading to find out:

• The basics of trigonometry;
• How to calculate a right triangle with trigonometry;
• A worked example of how to use trigonometry to calculate a right triangle with steps;

And much more!

Trigonometry is a branch of mathematics that relates angles to the length of specific segments . We identify multiple trigonometric functions: sine, cosine, and tangent, for example. They all take an angle as their argument, returning the measure of a length associated with the angle itself. Using a trigonometric circle , we can identify some of the trigonometric functions and their relationship with angles.

As you can see from the picture, sine and cosine equal the projection of the radius on the axis, while the tangent lies outside the circle. If you look closely, you can identify a right triangle using the elements we introduced above: let's discover the relationship between trigonometric functions and this shape.

Consider an acute angle in the trigonometric circle above: notice how you can build a right triangle where:

• The radius is the hypotenuse; and
• The sine and cosine are the catheti of the triangle.

α \alpha α is one of the acute angles, while the right angle lies at the intersection of the catheti (sine and cosine)

Let this sink in for a moment: the length of the cathetus opposite from the angle α \alpha α is its sine , sin ⁡ ( α ) \sin(\alpha) sin ( α ) ! You just found an easy and quick way to calculate the angles and sides of a right triangle using trigonometry.

The complete relationships between angles and sides of a right triangle need to contain a scaling factor, usually the radius (the hypotenuse). Identify the opposite and adjacent . We can then write:

By switching the roles of the legs, you can find the values of the trigonometric functions for the other angle.

Taking the inverse of the trigonometric functions , you can find the values of the acute angles in any right triangle.

Using the three equations above and a combination of sides, angles, or other quantities, you can solve any right triangle . The cases we implemented in our calculator are:

• Solving the triangle knowing two sides ;
• Solving the triangle knowing one angle and one side ; and
• Solving the triangle knowing the area and one side .

Take a right triangle with hypotenuse c = 5 c = 5 c = 5 and an angle α = 38 ° \alpha=38\degree α = 38° . Surprisingly enough, this is enough data to fully solve the right triangle! Follow these steps:

• Calculate the third angle: β = 90 ° − α \beta = 90\degree - \alpha β = 90° − α .
• sin ⁡ ( α ) = 0.61567 \sin(\alpha) = 0.61567 sin ( α ) = 0.61567 .
• o p p o s i t e = sin ⁡ ( α ) ⋅ h y p o t e n u s e = 0.61567 ⋅ 5 = 3.078 \mathrm{opposite} = \sin(\alpha)\cdot\mathrm{hypotenuse} = 0.61567 \cdot 5 = 3.078 opposite = sin ( α ) ⋅ hypotenuse = 0.61567 ⋅ 5 = 3.078 .
• a d j a c e n t = 0.788 ⋅ 5 = 3.94 \mathrm{adjacent} = 0.788\cdot 5 = 3.94 adjacent = 0.788 ⋅ 5 = 3.94 .

If you liked our right triangle trigonometry calculator, why not try our other related tools? Here they are:

• The trigonometry calculator ;
• The cosine triangle calculator ;
• The sine triangle calculator ;
• The trig triangle calculator ;
• The trig calculator ;
• The sine cosine tangent calculator ;
• The tangent ratio calculator ; and
• The tangent angle calculator .

How do I apply trigonometry to a right triangle?

To apply trigonometry to a right triangle, remember that sine and cosine correspond to the legs of a right triangle . To solve a right triangle using trigonometry:

• sin(α) = opposite/hypotenuse ; and
• cos(α) = adjacent/hypotenuse .
• By taking the inverse trigonometric functions , we can find the value of the angle α .
• You can repeat the procedure for the other angle.

What is the hypotenuse of a triangle with α = 30° and opposite leg a = 3?

The length of the hypotenuse is 6 . To find this result:

• Calculate the sine of α : sin(α) = sin(30°) = 1/2 .
• Apply the following formula: sin(α) = opposite/hypotenuse hypotenuse = opposite/sin(α) = 3 · 2 = 6 .

Can I apply right-triangle trigonometric rules in a non-right triangle?

Not directly: to apply the relationships between trigonometric functions and sides of a triangle, divide the shape alongside one of the heights lying inside it. This way, you can split the triangle into two right triangles and, with the right combination of data, solve it!

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Triangle Calculator

Please provide 3 values including at least one side to the following 6 fields, and click the "Calculate" button. When radians are selected as the angle unit, it can take values such as pi/2, pi/4, etc.

A triangle is a polygon that has three vertices. A vertex is a point where two or more curves, lines, or edges meet; in the case of a triangle, the three vertices are joined by three line segments called edges. A triangle is usually referred to by its vertices. Hence, a triangle with vertices a, b, and c is typically denoted as Δabc. Furthermore, triangles tend to be described based on the length of their sides, as well as their internal angles. For example, a triangle in which all three sides have equal lengths is called an equilateral triangle while a triangle in which two sides have equal lengths is called isosceles. When none of the sides of a triangle have equal lengths, it is referred to as scalene, as depicted below.

Tick marks on the edge of a triangle are a common notation that reflects the length of the side, where the same number of ticks means equal length. Similar notation exists for the internal angles of a triangle, denoted by differing numbers of concentric arcs located at the triangle's vertices. As can be seen from the triangles above, the length and internal angles of a triangle are directly related, so it makes sense that an equilateral triangle has three equal internal angles, and three equal length sides. Note that the triangle provided in the calculator is not shown to scale; while it looks equilateral (and has angle markings that typically would be read as equal), it is not necessarily equilateral and is simply a representation of a triangle. When actual values are entered, the calculator output will reflect what the shape of the input triangle should look like.

Triangles classified based on their internal angles fall into two categories: right or oblique. A right triangle is a triangle in which one of the angles is 90°, and is denoted by two line segments forming a square at the vertex constituting the right angle. The longest edge of a right triangle, which is the edge opposite the right angle, is called the hypotenuse. Any triangle that is not a right triangle is classified as an oblique triangle and can either be obtuse or acute. In an obtuse triangle, one of the angles of the triangle is greater than 90°, while in an acute triangle, all of the angles are less than 90°, as shown below.

Triangle facts, theorems, and laws

• It is not possible for a triangle to have more than one vertex with internal angle greater than or equal to 90°, or it would no longer be a triangle.
• The interior angles of a triangle always add up to 180° while the exterior angles of a triangle are equal to the sum of the two interior angles that are not adjacent to it. Another way to calculate the exterior angle of a triangle is to subtract the angle of the vertex of interest from 180°.
• The sum of the lengths of any two sides of a triangle is always larger than the length of the third side

a 2 + b 2 = c 2

EX: Given a = 3, c = 5, find b:

3 2 + b 2 = 5 2 9 + b 2 = 25 b 2 = 16 b = 4

• Law of sines: the ratio of the length of a side of a triangle to the sine of its opposite angle is constant. Using the law of sines makes it possible to find unknown angles and sides of a triangle given enough information. Where sides a, b, c, and angles A, B, C are as depicted in the above calculator, the law of sines can be written as shown below. Thus, if b, B and C are known, it is possible to find c by relating b/sin(B) and c/sin(C). Note that there exist cases when a triangle meets certain conditions, where two different triangle configurations are possible given the same set of data.
• Given the lengths of all three sides of any triangle, each angle can be calculated using the following equation. Refer to the triangle above, assuming that a, b, and c are known values.

Area of a Triangle

There are multiple different equations for calculating the area of a triangle, dependent on what information is known. Likely the most commonly known equation for calculating the area of a triangle involves its base, b , and height, h . The "base" refers to any side of the triangle where the height is represented by the length of the line segment drawn from the vertex opposite the base, to a point on the base that forms a perpendicular.

Another method for calculating the area of a triangle uses Heron's formula. Unlike the previous equations, Heron's formula does not require an arbitrary choice of a side as a base, or a vertex as an origin. However, it does require that the lengths of the three sides are known. Again, in reference to the triangle provided in the calculator, if a = 3, b = 4, and c = 5:

Median, inradius, and circumradius

The median of a triangle is defined as the length of a line segment that extends from a vertex of the triangle to the midpoint of the opposing side. A triangle can have three medians, all of which will intersect at the centroid (the arithmetic mean position of all the points in the triangle) of the triangle. Refer to the figure provided below for clarification.

The medians of the triangle are represented by the line segments m a , m b , and m c . The length of each median can be calculated as follows:

Where a, b, and c represent the length of the side of the triangle as shown in the figure above.

As an example, given that a=2, b=3, and c=4, the median m a can be calculated as follows:

The inradius is the radius of the largest circle that will fit inside the given polygon, in this case, a triangle. The inradius is perpendicular to each side of the polygon. In a triangle, the inradius can be determined by constructing two angle bisectors to determine the incenter of the triangle. The inradius is the perpendicular distance between the incenter and one of the sides of the triangle. Any side of the triangle can be used as long as the perpendicular distance between the side and the incenter is determined, since the incenter, by definition, is equidistant from each side of the triangle.

For the purposes of this calculator, the inradius is calculated using the area (Area) and semiperimeter (s) of the triangle along with the following formulas:

where a, b, and c are the sides of the triangle

Circumradius

The circumradius is defined as the radius of a circle that passes through all the vertices of a polygon, in this case, a triangle. The center of this circle, where all the perpendicular bisectors of each side of the triangle meet, is the circumcenter of the triangle, and is the point from which the circumradius is measured. The circumcenter of the triangle does not necessarily have to be within the triangle. It is worth noting that all triangles have a circumcircle (circle that passes through each vertex), and therefore a circumradius.

For the purposes of this calculator, the circumradius is calculated using the following formula:

Where a is a side of the triangle, and A is the angle opposite of side a

Although side a and angle A are being used, any of the sides and their respective opposite angles can be used in the formula.