Common Core Geometry Unit 3 Starting Points
Unit 3: Circles, Proofs, and Constructions
Essential Questions:
● Why are all circles similar?
● How are segments within circles, such as radii, diameters, and chords, related to
each other? What is the relationship of their measurements?
● How do inscribed, circumscribed, and central angles relate to each other?
● How do the constructions of inscribed and circumscribed circles relate to the
concurrencies explored during Unit 2?
● How can various figures be inscribed in a circle using various tools? How do the
properties of these figures relate to the parts of a circle?
● Why does the formula for the circumference and area of a circle work?
● How does the circumference of a circle relate to an intercepted arc? How can this
be used to define a radian?
● How does the area of a circle relate to the area of a sector? Can this be used to
derive a formula?
● How can coordinates be used to derive the equation for a circle given center and
radius? How can you determine whether or not a given point lies on this circle?
Curriculum Standards:
Understand and apply theorems about circles.
G.C.A.1 Prove that all circles are similar.
Understand and apply theorems about circles.
G.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.
Understand and apply theorems about circles.
G.C.A.3 Construct the inscribed and circumscribed circles of a triangle, and prove
properties of angles for a quadrilateral inscribed in a circle.
Make geometric constructions.
G.CO.C.13 Construct an equilateral triangle, a square, and a regular hexagon inscribed in
a circle.
Understand and apply theorems about circles.
G.C.A.4 (+) Construct a tangent line from a point outside a given circle to the circle.
Explain volume formulas and use them to solve problems.
G.GMD.A.1 Give an informal argument for the formulas for the circumference of a
circle, area of a circle. Use dissection arguments and informal limit arguments.
This document represents one sample starting points for the unit. It is not all-inclusive and is only
one planning tool. Please refer to the wiki for more information and resources.
Find arc lengths of sectors of circles.
G.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.
Translate between the geometric description and the equation for a conic section.
G.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.
Use coordinates to prove simple geometric theorems algebraically.
G.GPE.B.4 Use coordinates to prove simple geometric theorems algebraically. i.e. prove
or disprove that the point lies on the circle centered at the origin and containing the
point (0, 2).
Approximate Length: 35-40 days (30-35 G/T)
Standard(s)
Days
Notes
G.C.A.1
Big Ideas:
1-2
All circles are similar, which makes them
proportional. This can be verified using
transformations.
Resources:
 Lesson: Similar Circles
G.C.A.2
G.C.A.4 (+)
15-18
Big Ideas:
Diameters, radii, chords, tangents and secants
are all parts of a circle.
The parts of a circle are related but have their
own unique properties. Based on these
properties and relationships, lengths can be
calculated.
Angles formed by parts of a circle are related
and measurements can be calculated based on
these relationships.
Resources:
 Lesson Seed: Parts of a Circle
 Lesson: Angles in a Circle
 Lesson: Inscribed and Central Angles
This document represents one sample starting points for the unit. It is not all-inclusive and is only
one planning tool. Please refer to the wiki for more information and resources.
 Lesson: Tangent Lines
 Sketchpad Tutorial: Inscribed Angles
 Task: Getting a Job (G.C.A.4+)
Assessment Items:
 Illustrative Mathematics: Tangent Lines
and the Radius of a Circle
 Illustrative Mathematics: Neglecting the
Curvature of the Earth
G.C.A.3
G.CO.C.13
6-8
Big Ideas:
Circles can be inscribed or circumscribed with a
variety of tools.
Constructing inscribed regular polygons in
circles can reinforce properties of those
polygons.
Resources:
 Task: Your Trip to Paris
 Sketchpad Tutorials: Inscribing and
Circumscribing Circles
 Compass Tutorials: Inscribing and
Circumscribing Circles
 Lesson: Constructing a Square Inscribed
in a Circle
 Lesson: Inscribing Figures
 Construction Tutorial: Square Inscribed
in a Circle (Paper Plate)
Assessment Items:
 PARCC Prototype: Geometric
Construction Connection
 Illustrative Mathematics: Inscribing a
Circle in a Triangle II
 Illustrative Mathematics: Right Triangles
Inscribed in Circles II
 Illustrative Mathematics: Inscribing a
Square in a Circle
 Illustrative Mathematics: Inscribing a
Hexagon in a Circle
G.MD.A.1
2-3
Big Ideas:
The formula for the area of a circle can be
This document represents one sample starting points for the unit. It is not all-inclusive and is only
one planning tool. Please refer to the wiki for more information and resources.
argues using dissection and limits.
The formula for the circumference of a circle
can be investigated using proportionality.
Resources:
 Lesson: Informal Proofs of
Circumference and Area of a Circle
 Construction Tutorial: Approximating
Area of a Circle
G.C.B.5
6-8
Assessment Items:

Big Ideas:
Arc length is proportional to the radius of a
circle. A radian is a constant of proportionality.
The formula of the area of a sector can be
derived through proportionality.
Real-world problems can be solved using arc
length and the area of sectors.
Resources:
 Task: How on Earth
 Lesson: Measures of Arcs and Radians
 Lesson: Area of a Sector

Assessment Items:
 Illustrative Mathematics: Eratosthenes
and the circumference of the earth
 Illustrative Mathematics: Setting Up
Sprinklers
This document represents one sample starting points for the unit. It is not all-inclusive and is only
one planning tool. Please refer to the wiki for more information and resources.
G.GPE.A.1
G.GPE.B.4
5-7
Big Ideas:
The equation of a circle can be derived using the
Pythagorean theorem.
The general equation of a circle can be found
given the center and radius.
The equation of a circle can be found given a
coordinate and the radius and a center at the
origin.
Using coordinates, it can be verified that given
points lie on or off a circle with given
constraints.
Resources:
 Task: Cruisin’ without Communication
 Lesson: Equation of a Circle
 Lesson Seed: Proofs with Circles in the
Coordinate Plane
Assessment Items:
 Illustrative Mathematics: Triangles
Inscribed in a Circle
Howard County Public Schools Office of Secondary Mathematics Curricular Projects has
licensed this product under a Creative Commons Attribution-NonCommercial-NoDerivs
3.0 Unported License.
This document represents one sample starting points for the unit. It is not all-inclusive and is only
one planning tool. Please refer to the wiki for more information and resources.