equations of circles common core geometry homework

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Common Core High School Geometry Teaching and Learning Resources

Domain: c: circles.

Select the cluster for resources on this page:

C-A : Understand and apply theorems about circles

C-B : Find arc lengths and areas of sectors of circles

C-A: Understand and apply theorems about circles

Standards :

• C-A.1. Prove that all circles are similar.
• C-A.2. Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle.
• C-A.3. Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle.
• C-A.4. (+) Construct a tangent line from a point outside a given circle to the circle.

Technology-enhanced investigations:

Mathwords.com: Math Dictionary : Key vocabulary for this domain.  Use with C-A and C-B:

CK-12:Geometry: concepts, videos, and practice exercises for learners to complete: Use with C-A.2.

• Inscribed Angles in Circles
• Tangent Lines : Lines perpendicular to the radius drawn to the point of tangency.
• Chords : Arcs determined by angles whose vertex is the center of a circle and chords (segments that connect two points on a circle).
• Angles On and Inside a Circle
• Angles Outside a Circle

Khan Academy: Geometry: Circles :

• Proof: all circles are similar : video.  Aligns with C-A.1.

Khan Academy: Geometric Constructions , Tutorial videos:

• Construct a circle inscribed and circumscribed about a triangle : Aligns with C-A.3.
• Constructing a line tangent to a circle : 2 videos: given P on the circle, and P outside of the circle.  Aligns with C-A.4.

Imagine Learning Classroom (formerly LearnZillion) subscription needed:

• Lesson set: Prove that all circles are similar : Four video lessons: Similarity is demonstrated in these using similar triangles, translations and dilations.  Aligns with C-A.1.
• Lesson set: Identify and describe relationships among inscribed angles, radii, and chords : Four video lessons: Find measures of inscribed angles using central angles, understand tangents are perpendicular to radii, find relationships between central angles and circumscribed angles, find missing angle measurements within a circle using the intersecting chord theorem.  Aligns with C-A.2.
• Lesson set: Construct the inscribed and circumscribed circles of a triangle; prove properties of angles for a quadrilateral inscribed in a circle : Four video lessons: Construct circumscribed, inscribed circle of a triangle; find circumcenters and incenters of a triangle; show angle relationships of cyclic quadrilaterals by using Central Angle Theorem.  Aligns with C-A.3.

MathsIsFun.com: Constructions : This is a series of 17 animated geometric constructions, which learners would do with a compass, pencil, and straightedge only.  Among those are circle constructions to find a center, tangent lines from an external point, inscribed and circumscribed circles for triangles, circle touching three points, and a pentagon construction.  Aligns with C-A.3 and C-A.4.

Math Open Reference: Constructions : The site provides step-by-step animations of constructions with compass and straightedge.  The list is comprehensive for lines, line segments, angles, triangles, circles, arcs, ellipses, polygons, and non-Euclidean constructions.  The following align with C-A.3 and C-A.4:

• Incircle of a triangle
• Circumcircle of a triangle
• Tangent to a circle at a point
• Tangents through an external point

NCTM Illuminations:

• Incenter-Incircle : This applet allows you to find the incenter and incircle of a triangle, and provides a visual explanation of why it occurs where it does.
• Circumcircle : Applet to construct the circumcircle about a triangle.
• Simson Line : Applet to construct and explore the Simson line.

Thinking Mathematics: Tangent Theorems for Circles and the Pythagorean Theorem : A YouTube video on two results from the geometry of circles that are easily proven using the Pythagorean theorem.

Wolfram Demonstrations Project: Download the free Wolfram CDF player to interact with the following manipulatives.  Note: Within the Wolfram Demonstration Project are 21 manipulatives addressing C-A.2 , 10 manipulatives for C-A.3 , and 1 manipulative for C-A.4 .  Among those:

• Geometric elements of a circle : Aligns with C-A.2.
• Inscribed and central angles of a circle : Aligns with C-A.2.
• Inscribed angles that intercept the same arc : Aligns with C-A.2.
• Inscribed angles in a circle subtended by the same chord : Aligns with C-A.2.
• Thales' theorem: angles inscribed in a semi-circle : Aligns with C-A.2.
• Thales' theorem: a vector-based proof : Aligns with C-A.2.
• Equally distance chords : Aligns with C-A.2.
• Tangent chord angle : Aligns with C-A.2.
• Intersecting secants theorem : Aligns with C-A.2.
• Intersecting chords theorem : Aligns with C-A.2.
• The Chordal theorem : Aligns with C-A.2.
• The perpendicular bisector of a chord : Aligns with C-A.2.
• Sum of Opposite angles of a quadrilateral inscribed in a circle is 180 degrees : Aligns with C-A.2.
• Euclid, Book 3, Proposition 22 : in a cyclic quadrilateral, opposite angles sum to 180 degrees: Aligns with C-A.2, C-A.3, and CO-C.10.
• Circles and triangles : Aligns with C-A.2.
• Three points determine a circle : Aligns with C-A.2.
• Reflection in a circle : Aligns with C-A.2.
• Tangent circles : Aligns with C-A.2.
• The bomb problem : This problem considers multiple points on a plane.  In the "bomb problem", the smallest circle that contains all the points is found.  The bomb problem comes from military tactics. What is the least powerful bomb that can destroy all targets, and where should it be dropped?  Aligns with C-A.2, C-A.3.
• Triangle Calculator : The program calculates triangle centers: incenter I, centroid G, circumcenter O, orthocenter H.  The points G, O, and H are collinear: Aligns with C-A.3.
• Problems on Circles IX: Circumcircles and Incircles of a Triangle : Aligns with C-A.3.
• Circumcircle and incircle of a triangle : Aligns with C-A.3.
• Johnson's theorem : Let three circles of equal diameter intersect at a point H and intersect pairwise at points A, B, and C. Then the circumcircle of the triangle ABC has the same diameter as the other circles.  Aligns with C-A.3.
• Pairwise Tangent Circles Centered at the Vertices of a Triangle : Aligns with C-A.3.
• Tangents to a circle : Aligns with C-A.2, C-A.4.

Multiple Choice:

MathsIsFun.com: Circle Theorems : Content reviews inscribed angle theorems, angles in quadrilaterals inscribed in circles, the 90 degree angle between a tangent line and the radius drawn to the point of tangency.  Examples are shown.  Learners then complete 10 multiple choice problems on those theorems.  Aligns with C-A.2 and C-A.3.

Constructed-response:

Khan Academy: HSG-G-C includes HSG-C.A.1 through HSG-C.A.4 Common Core aligned problems.

Also see Geometry: Circles : Practice questions with videos:

• Quiz: Inscribed Quadrilaterals : Aligns with C-A.2, C-A.3.
• Tangents of Circles Problems : involves central, inscribed, and circumscribed angles.  Aligns with C-A.4.

Performance tasks:

Illustrative Mathematics: Geometry :

• Similar circles : Aligns with C-A.1.
• Right triangles inscribed in circles 1 : Aligns with C-A.2.
• Inscribing a circle in a triangle 1 : Aligns with C-A.3.

Inside Mathematics: MARS Tasks:

• Circle and Squares : Students calculate sides of squares given the radius of the inscribed and circumscribed circle and calculate ratios of nested circles and squares.  This task aligns with standards C-A.2 and C-A.3.
• Circles in Triangles : Students use geometric properties of circles and triangles to prove that two triangles are congruent.  This task aligns with geometry congruence standards CO-C.9 and CO-C.10 and geometry circle standards C-A.2 and C-A.3.

Mathematics Assessment Project: High School: Geometry:

• Task 403: Inscribing and Circumscribing Right Triangles
• Task 222: Geometry Problems: Circles and Triangles

Mathematics Vision Project, Secondary 2 Student Edition :

• Module 7: Circles a Geometric Perspective : This module contains 10 classroom tasks.  Standards C-A.1, C-A.2, C-A.3, and C-A.4 can be found within tasks 1-3 and task 6.

Back to top

C-B: Find arc lengths and areas of sectors of circles

• C-B.5. Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector.

CK-12: Geometry: Arcs in Circles : concepts, videos, and practice problems for learners to complete.

Thinking Mathematics: Radian Measure : A YouTube video with explanation of the radian measure.

Wolfram Demonstrations Project: Download the free Wolfram CDF player to interact with the following manipulatives.  Note: Within the Wolfram Demonstration Project are 10 manipulatives for C-B.5 .  Among those:

• Angles measured in degrees and radians
• Angular displacement : arc length

MathsIsFun.com: Circle Sector and Segment : Content reviews arc length and area of a sector showing how formulas are derived.  It also includes how to find the area of a segment.  Learners can complete 10 multiple choice problems.  Use with C-B.

Khan Academy: HSG-G-C includes HSG-C.B.5 Common Core aligned problems.

Also see Geometry: Circles Practice questions with videos:

• Arc measures in degrees : Aligns with C-B.5.
• Arc Length : Relate the length of the arc to the circumference and the central angle subtended by the arc in degrees.  Aligns with C-B.5.
• Radians and Degrees : Convert angle measures from degrees to radians and vice versa.  Aligns with C-B.5.
• Radians and Arc Length : Solve problems related to radians and arc length, like finding an arc length given a central angle in radians and a radius.
• Area of a sector : Relate the area of a sector to the area of the whole circle.  Aligns with C-B.5.

Illustrative Mathematics: Geometry : Setting up sprinklers : This task aligns with standards in C-B.

Mathematics Assessment Project: High School: Geometry: Task 441: Sectors of Circles

• Module 7: Circles a Geometric Perspective : This module contains 10 classroom tasks.  Standard C-B.5 can be found within tasks 7-9.

Common Core Math: Intro | HS Geometry Domain: CO   |   SRT   |   C   |   GPE   |   GMD   |   MG   |

Contact Dr. Patricia Deubel: [email protected]

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Deubel, P. (year from last revised).  Common Core Math: High school geometry teaching and learning resources . Computing Technology for Math Excellence. Retrieved Month, Day, Year, from https://www.ct4me.net/Common-Core/hsgeometry/hsg-circles.htm

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How to Teach Circles Using the Common Core Standards?

Here is a great video to assist you in starting your lessons on circles. We all know the standards have changed and this one specifically is a head ache due to the algebra 2 that is involved.

CCSS.Math.Content.HSG-GPE.A.1  Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation.

Most of us have students that have not learned how to complete the square or even mastered basic algebraic proofs. Deriving the equation of a circle is impossible without being able to complete the square. Below is a great video on how to teach them this algebra lesson so you can teach this geometry standard.

Once you have reviewed / taught them completing the square. You then have to go into much more complex problems to get them to this level for the standard.

Here is your free content for this lesson!

Circles in the Coordinate Plane Worksheet – Word Docs & PowerPoints

To gain access to our editable content Join the Geometry Teacher Community! Here you will find hundreds of lessons, a community of teachers for support, and materials that are always up to date with the latest standards.

12-5 Assignment Student Edition – Circles in the Coordinate Plane  ( FREE ) 12-5 Assignment Teacher Edition – ( Members Only ) 12-5 Bell Work Student Edition – Circles in the Coordinate Plane  ( FREE ) 12-5 Bell Work Teacher Edition – ( Members Only ) 12-5 Exit Quiz Student Edition – Circles in the Coordinate Plane  ( FREE ) 12-5 Exit Quiz Teacher Edition – ( Members Only ) 12-5 Guide Notes Student Edition – Circles in the Coordinate Plane  ( FREE ) 12-5 Guided Notes Teacher Edition – ( Members Only ) 12-5 Slide Show – Circles in the Coordinate Plane  ( FREE ) 12-5 Lesson Plan – ( Members Only ) 12-5 Online Activities – Circles in the Coordinate Plane ( Members Only )

Hopefully this will assist you with getting your class to master this standard for 10th Grade Geometry. If you would like to see all of our materials on this standard simply join the Geometry Teacher Community by clicking the image below. Everything you will need to master the Common Core Standards in your Geometry Class will be given to you!

Click Here to Get Access Now!

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Examples for

Common Core Math: High School Geometry: Circles

In high school, students define and describe measurements of circles, using terms like radius, diameter, chord and tangent. Students compute arc length and sector area and apply theorems involving central, inscribed and circumscribed angles. Students also use angles to analyze circles and their relationships to other shapes, including by constructing inscribed and circumscribed circles of triangles and quadrilaterals.

Common Core Standards

Get information about Common Core Standards.

Look up a specific standard:

Search for all standards in a domain:, inscribed & circumscribed circles.

Relate circles to other geometric figures.

Inscribe circles within polygons (CCSS.Math.Content.HSG-C.A.3):

Find circumcircles surrounding polygons (ccss.math.content.hsg-c.a.3):, measurements in circles.

Define and compute measurements and properties of circles.

Prove that all circles are similar (CCSS.Math.Content.HSG-C.A.1):

Recognize and apply properties of angles in circles (ccss.math.content.hsg-c.a.2):, find lines tangent to a circle (ccss.math.content.hsg-c.a.4):, compute arc lengths and sector areas (ccss.math.content.hsg-c.b.5):, related examples.

• Common Core Math: Congruence
• Common Core Math: Expressing Geometric Properties with Equations
• Common Core Math: Measurement & Dimension
• Common Core Math: Modeling with Geometry
• Common Core Math: Similarity & Trigonometry

Chapter 10, Lesson 8: Equations of Circles

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Common Core Geometry Unit 1 Lesson 5 - Circles and Arcs

emathinstruction

Aug 17, 2018

Learning the Common Core Geometry Unit 1 Lesson 5 - Circles and Arcs by eMathInstruction

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Common Core Geometry – by Kirk

So, the summer has been a great mix of working on Add-Ons for the three courses we currently offer and thinking a lot about Common Core Geometry. I’ve now completed the first four units and a rough course outline that’s been indexed to the CCSS Standards for CC Geometry (as defined by PARCC and refined by NYSED).

I’ve got all of the lessons for those four units posted at this point under our Courses section:

You can find the course outline if you click on the Table of Contents and Standards Documents link. All of the lessons and homework sets posted for the first four units are in rough draft form only. All diagrams that occur in those files were created on our new Efofex software.

The units of the course (at least at this point) are:

Unit 1 – Essential Geometric Tools and Concepts

Unit 2 – Transformations, Rigid Motions, and Congruence

Unit 3 – Euclidean Triangle Proofu

Unit 4 – Constructions

Unit 5 – The Tools of Coordinate Geometry

Unit 6 – Quadrilaterals

Unit 7 – Dilations and Similarity

Unit 8 – Right Triangle Trigonometry

Unit 9 – Circles with and without Coordinates

Unit 10 – The Geometry of Three Dimensions

I don’t think the order of the units is anything too radical, but I wanted to discuss why I have them in this order and my overall thinking at this point. I do want to be clear up front that I believe geometry is the most tactile of all mathematical fields. It is about space and that deserves not just proof and algebraic problems, but also the tools of geometry, which include compass, ruler, straightedge, protractor, and, yes, tracing paper. We currently developing a really good tracing paper that we will begin to sell on our site right after school begins.

Geometry is really all about exploring physical space using tools we develop in mathematics, whether that be transformations, Euclidean logic, or Cartesian coordinates. I began the course in Unit 1 by making sure that students have some of the very basics down, including a really good sense for circles and arcs of circles.

In Unit 2, I wanted to establish certain properties about space by using H. Wu’s rigid motion work with congruence. This is an awesome unit, but probably one of the newer ones in the Common Core. Within this unit, though, we use rigid motions to establish a number of important facts, such as SAS, ASA, and SSS criteria for triangle congruence, facts about the perpendicular bisector of a segment, and parallel line properties.

Unit 3 puts us back on more solid ground with Euclidean triangle geometry proof. I did bring parallel lines back into the mix so that students could also be exposed to AAS and HL methods of proving triangles congruent. We look at many of the classic problems of geometry in this unit, including why the angle bisector represents all points equidistant from the sides of an angle.

In Unit 4 we explore the beautiful world of constructions. I know constructions can be a mystery to students, a set of directions that they need to memorize to carry out the construction. I make sure within every lesson that the constructions have both purpose and are proven by using Euclidean triangle proof.

I haven’t created the remaining units, but want to explain how they will unfold (hopefully). In Unit 5, I am going to introduce coordinate geometry. I wanted to do this relatively early in the course for two primary reasons: (1) so students could have a break from the world of Euclidean reasoning and (2) so that we could use coordinate geometry as well as Euclidean geometry and Wu’s transformation work to explore quadrilaterals (Unit 6) and similarity (Unit 7).

Of course, once we’ve established the concept of similarity in Unit 7, we will naturally move to right triangle trigonometry in Unit 8. Right triangle trig is one of the most applied fields within geometry and we will make sure students understand its basis in similarity and then its application in the real world.

Unit 9, which will come towards the end of the third marking period, will challenge students will the geometry of the circle. But, at this point, we will have a good background in congruence, similarity, the distance formula, and other tools that will allows students to explore the geometry of circles in the Euclidean and Cartesian planes.

Finally, Unit 10 will concentrate on three dimensional geometry as well as measurement and geometric modeling.

I will continue to work on lessons (Unit 5 here I come) for the remainder of the summer and will then begin video work for the first few units once Labor Day is behind us and my own kids have gone back to school (Max and Evie). Hopefully by late September all videos will be done for Units 1 through 4.

Work will continue on the units with pauses in October and November for all sorts of conferences (AMTNYS, AMTNYC, etc). I think, given my current pace, that the first draft of all units should be done by Winter break. I would then expect videos and answer keys to be done with books ready to order by late winter or early spring.

Feel free to give me feedback if you try the materials. If you have questions about the order or any given lesson, just shoot me an email: [email protected]

7 thoughts on “ Common Core Geometry – by Kirk ”

I was an avid user of your material for Alg2/Trig. I love and appreciate your work. I’m teaching Geometry this year and I’m so grateful you took on this endeavor. I’m new to the curriculum and would love to follow what you have with the Units. My only issue is timing. How long into the school year do you think the current 4 units would get me, and would units 5/6 or even 7 be available before I finish the first 4? I just don’t want to follow along with what you have for 4 units and then not have anything to continue with. In my opinion noting out there compares to the product you put out. I can’t wait till next year when all is done and I can get my school district to purchase everything from you like I did for Alg2/Trig. Thanks in advance for your response.

Michael. The first four units have about 40 lessons in them. So, I would assume that would get you easily through the first marking period and into a bit of the second, given testing and all the daily disruptions. I will certainly, certainly finish Units 5 and 6 long before you finish Unit 4. I’m working on 5 right now and am in the middle of it, but other work has been keeping me busy so far this year. I do hope though to have the entire curriculum done by Winter break.

By Winter Break you mean in February? That’s great that extra units will be up before we finish unit 4. I seriously cannot thank you enough for all you have done to help all of us teachers struggling with the constant changes to the curriculum. You’re a lifesaver!

I’m hoping by late December to be done.

Are there answer keys and assessment material available for the Geometry course for purchase or shared anywhere?

Mike. Not yet. We hope to have the answer keys available in late winter/early spring. Sorry about that. If you have any questions about topics or lessons in general, just send me an email, [email protected] .

I am anxiously awaiting this product to be completed! Love the Algebra 1 and 2. I have just taken over Geometry part way through the year, due to an unfortunate circumstance, and have never taught Geometry. I can not wait until this is full release and I can have the tests/review/answer keys! 🙂

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Common Core Math In The Classroom And Homework Help

How can the Common Core Math be implemented in the Classroom? How can I teach the Common Core Math at home? How can I get homework help for the Common Core Math?

Related Pages Common Core Math Video Lessons, Math Worksheets & Games for all grades Math Worksheets

The following lessons are based on the New York State (NYS) Common Core Math Standards. They consist of lesson plans, worksheets (from the NYSED) and videos to help you prepare to teach Common Core Math in the classroom or at home. There are lots of help for classwork and homework.

Each grade is divided into six or seven modules. Mid-module and End-Module Assessments are also included.

The lessons are divided into Fluency Practice, Application Problem, Concept Development, and Student Debrief. The worksheets are divided into Problem Set, Exit Ticket, and Homework.

Kindergarten Mathematics Numbers to 10 Two-Dimensional and Three-Dimensional Shapes Comparison of Length, Weight, Capacity Number Pairs, Addition and Subtraction to 10 Counting to 100 Analyzing, Comparing, and Composing Shapes Grade 1 Mathematics Sums and Differences to 10 Introduction to Place Value Through Addition and Subtraction Within 20 Ordering and Comparing Length Measurements as Numbers Place Value, Comparison, Addition and Subtraction to 40 Identifying, Composing, and Partitioning Shapes Place Value, Comparison, Addition and Subtraction to 100 Grade 2 Mathematics Sums and Differences to 20 Addition and Subtraction of Length Units Place Value, Counting, and Comparison of Numbers to 1,000 Addition and Subtraction Within 200 with Word Problems to 100 Addition and Subtraction Within 1,000 with Word Problems to 100 Foundations of Multiplication and Division Problem Solving with Length, Money, and Data Fractions as Equal Parts of Shapes, Time Grade 3 Mathematics Properties of Multiplication and Division and Solving Problems with Units of 2 and 10 Place Value and Problem Solving with Units of Measure Multiplication and Division with Units of 0, 1, 6, and Multiples of 10 Multiplication and Area Fractions as Numbers on the Number Line Collecting and Displaying Data Geometry and Measurement Word Problems Grade 4 Mathematics Place Value, Rounding, and Algorithms for Addition and Subtraction Unit Conversions and Problem Solving with Metric Measurement Multi-Digit Multiplication and Division Angle Measure and Plane Figures Fraction Equivalence, Ordering, and Operations Decimal Fractions Exploring Measurement with Multiplication Grade 5 Mathematics Place Value and Decimal Fractions Multi-Digit Whole Number and Decimal Fraction Operations Addition and Subtraction of Fractions Line Plots of Fraction Measurements Addition and Multiplication with Volume and Area Problem Solving with the Coordinate Plane Grade 6 Mathematics Ratios and Unit Rates Arithmetic Operations Including Division of Fractions Rational Numbers Expressions and Equations Area, Surface Area, and Volume Problems Statistics Grade 7 Mathematics Ratios and Proportional Relationship Rational Numbers Expressions and Equations Percent and Proportional Relationships Statistics and Probability Geometry Grade 8 Mathematics Integer Exponents and Scientific Notation The Concept of Congruence Similarity Linear Equations Examples of Functions from Geometry Linear Functions Introduction to Irrational Numbers Using Geometry High School Algebra I Linear and Exponential Sequences Functions and Their Graphs Transformations of Functions Using Functions and Graphs to Solve Problems High School Geometry Congruence, Proof, and Constructions Similarity, Proof, and Trigonometry Extending to Three Dimensions Connecting Algebra and Geometry Through Coordinates Circles With and Without Coordinates

Have a look at the following videos for insights on how to implement the Core in classrooms and homes across America. We also have lesson plans, assessments and worksheets to help you in your preparation.

In this first video, we will join Sarah as she explains the Common Core State Standards and offers insights on how to implement the Core in classrooms. We will learn how teachers and students can shift their math classrooms to promote mathematical reasoning.

She emphasized on the need to focus on fewer concepts, coherence for mastery and an approach with more rigor. Focus means less rote memorization and more deep procedural knowledge and conceptual understanding. Rigor means having procedural fluency and conceptual understanding.

She talks about the six shifts in teaching Mathematics: Focus, Coherence, Fluency, Deep Understanding, Application, and Dual Intensity. Classrooms should be creative, engaged and even noisy. Families can be involved in applying the mathematical concepts.

In this video Sarah explains what is the Common Core and where did it come from. How to read the Common Core State Standards with confidence and perspective.

She explains that “Common doesn’t mean the same and the Standards are not the curriculum”. She also shows how to read the grade level standards for mathematics and for reading and writing.

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