Standard 6: Use properties and theorems related to circles.

General Information
Number: MA.912.GR.6
Title: Use properties and theorems related to circles.
Type: Standard
Subject: Mathematics (B.E.S.T.)
Strand: Geometric Reasoning

Related Benchmarks

This cluster includes the following benchmarks.

Related Access Points

This cluster includes the following access points.

Access Points

MA.912.GR.6.AP.1
Identify and describe the relationship involving the length of a secant, tangent, segment or chord in a given circle.
MA.912.GR.6.AP.2
Identify the relationship involving the measures of arcs and related angles, limited to central, inscribed and intersections
MA.912.GR.6.AP.3
Identify and describe the relationship involving triangles and quadrilaterals inscribed in a circle.
MA.912.GR.6.AP.4
Identify and describe the relationship involving the arc length and area of a sector in a given circle.

Related Resources

Vetted resources educators can use to teach the concepts and skills in this topic.

Formative Assessments

Central and Inscribed Angles:

Students are asked to describe the relationship between a central angle and an inscribed angle that intercept the same arc.

Type: Formative Assessment

Students are asked to draw a circle, a tangent to the circle, and a radius to the point of tangency. Students are then asked to describe the relationship between the radius and the tangent line.

Type: Formative Assessment

Softball Complex:

Students are asked to solve a design problem in which a softball complex is to be located on a given tract of land subject to a set of specifications.

Type: Formative Assessment

Students are asked to prove that opposite angles of a quadrilateral, inscribed in a circle, are supplementary.

Type: Formative Assessment

Deriving the Sector Area Formula:

Students are asked to write a formula to find the area of a sector of a circle and then explain and justify that formula.

Type: Formative Assessment

Students are asked to explain why the length of an arc intercepted by an angle is proportional to the radius and then explain how that proportionality leads to a definition of the radian measure of an angle.

Type: Formative Assessment

The Sprintersâ€™ Race:

Students are given a grid with three points (vertices of a right triangle) representing the starting locations of three sprinters in a race and are asked to determine the center of the finish circle, which is equidistant from each sprinter.

Type: Formative Assessment

Inscribed Angle on Diameter:

Students are asked to find the measures of two inscribed angles of a circle.

Type: Formative Assessment

Similar Circles:

Students are given two circles with different radii and are asked to prove that the circles are similar.

Type: Formative Assessment

Circles with Angles:

Students are given a diagram with inscribed, central, and circumscribed angles and are asked to identify each type of angle, determine angle measures, and describe relationships among them.

Type: Formative Assessment

All Circles Are Similar:

Students are given two circles with different radius lengths and are asked to prove that the circles are similar.

Type: Formative Assessment

Sector Area:

Students are asked to find the areas of sectors in two different circles.

Type: Formative Assessment

Arc Length:

Students are asked to find the lengths of arcs in two different circles.

Type: Formative Assessment

Lesson Plans

Why are Circles Similar?:

Type: Lesson Plan

The Seven Circles Water Fountain:

Students will apply concepts related to circles, angles, area, and circumference to a design situation.

Type: Lesson Plan

Are All Circles Similar?:

This lesson allows students to prove that all circles are similar using transformations. Students will need prior knowledge of similarity, transformations, and the definition of a circle. The lesson begins with a warm up regarding dilations, then poses the question: Are all circles similar? The students are guided through the proof using a translation and dilation. The teacher emphasizes the details in the proof. The lesson closes with an exit ticket.

Type: Lesson Plan

Congruence vs. Similarity:

Students will learn the difference between congruence and similarity of classes of figures (such as circles, parallelograms) in terms of the number of variable lengths in the class. A third category will allow not only rigid motions and dilations, but also a single one-dimensional stretch, allowing more classes of figures to share sufficient common features to belong.

Type: Lesson Plan

Circle to Circle:

Students use coordinate-based translations and dilations to prove circles are similar.

Type: Lesson Plan

Paper Plate Origami:

A hands-on activity where students construct inscribed regular polygons in a circle using models. Through guided questions, students will discover how to divide a model (paper plate) into 3, 4, and 6 parts. Using folding, a straightedge, and a compass, they will construct an equilateral triangle, a square, and a regular hexagon in their circles.

Type: Lesson Plan

My Favorite Slice:

The lesson introduces students to sectors of circles and illustrates ways to calculate their areas. The lesson uses pizzas to incorporate a real-world application for the of area of a sector. Students should already know the parts of a circle, how to find the circumference and area of a circle, how to find an arc length, and be familiar with ratios and percentages.

Type: Lesson Plan

Off on a Tangent:

Students learn and apply vocabulary, notation, concepts, and geometric construction techniques associated with circles and their tangents to a historical real-world scenario, the Mason-Dixon Line, and a hypothetical real-world scenario, the North-South Florida Line.

Type: Lesson Plan

Seeking Circle Angles:

Students will start this lesson with a win-lose-draw game to review circle vocabulary words. They will then use examples on a discovery sheet to discover the relationships between arcs and the angles whose vertex is located on a circle, in the interior of the circle, and exterior to the circle. They will wrap up the lesson in a class discussion and questions answered on white boards.

Type: Lesson Plan

Seeking Circle Segments:

Students will start this lesson with a "Pictionary" game to review circle vocabulary terms. They will then use computers and GeoGebra to discover the relationships between portions of segments that intersect in the interior of the circle, and exterior to the circle. They will wrap up the lesson in a class discussion and consensus on rules (formulas).

Type: Lesson Plan

Scientific calculations from a distant planet:

Students will act as mathematicians and scientists as they use models, observations and space science concepts to perform calculations and draw inferences regarding a fictional solar system with three planets in circular orbits around a sun. Among the calculations are estimates of the size of the home planet (using a method more than 2000 years old) and the relative distances of the planets from their sun.

Type: Lesson Plan

The Grass is Always Greener:

The lesson introduces area of sectors of circles then uses the areas of circles and sectors to approximate area of 2-D figures. The lesson culminates in using the area of circles and sectors of circles as spray patterns in the design of a sprinkler system between a house and the perimeter of the yard (2-D figure).

Type: Lesson Plan

In this lesson, students will convert from degrees to radians and radians to degrees and calculate arc length using both degrees and radians. Students will come to consensus as to why radians are the preferred measure of an angle. This lesson normally takes two 50 minute class periods to teach.

• Day 1: Bell Ringer-Day 1, Lesson Notes, Activity 1
• Day 2: Day 1 Review and Wrap-up, Bell Ringer-Day 2, Activity 2

Type: Lesson Plan

Perspectives Video: Professional/Enthusiast

All Circles Are Similar- Especially Circular Pizza!:

What better way to demonstrate that all circles are similar then to use pizzas! Gaines Street Pies explains how all pizza pies are similar through transformations.

Type: Perspectives Video: Professional/Enthusiast

The Lighthouse Problem:

This problem asks students to model phenomena on the surface of the earth by examining the visibility of the lamp in a lighthouse from a boat.

Solar Eclipse:

This problem solving task encourages students to explore why solar eclipses are rare by examining the radius of the sun and the furthest distance between the moon and the earth.

Eratosthenes and the circumference of the earth:

This problem solving task gives an interesting context for implementing ideas from geometry and trigonometry.

Triangles inscribed in a circle:

This problem solving task challenges students to use ideas about linear functions in order to determine when certain angles are right angles.

Inscribing a square in a circle:

This task provides an opportunity for students to apply triangle congruence theorems in an explicit, interesting context.

Two Wheels and a Belt:

This task combines two skills: making use of the relationship between a tangent segment to a circle and the radius touching that tangent segment, and computing lengths of circular arcs given the radii and central angles.

Right triangles inscribed in circles II:

Placing a Fire Hydrant:

This problem solving task asks students to place a fire hydrant so that it is equal distance from three given points.

Inscribing a triangle in a circle:

This problem introduces the circumcenter of a triangle and shows how it can be used to inscribe the triangle in a circle.

Circumcenter of a triangle:

This task shows that the three perpendicular bisectors of the sides of a triangle all meet in a point, using the characterization of the perpendicular bisector of a line segment as the set of points equidistant from the two ends of the segment.

Tangent Lines and the Radius of a Circle:

This problem solving task challenges students to find the perpendicular meeting point of a segment from the center of a circle and a tangent.

Setting Up Sprinklers:

This modeling task involves several different types of geometric knowledge and problem-solving: finding areas of sectors of circles, using trigonometric ratios to solve right triangles, and decomposing a complicated figure involving multiple circular arcs into parts whose areas can be found.

Neglecting the Curvature of the Earth:

This task applies geometric concepts, namely properties of tangents to circles and of right triangles, in a modeling situation. The key geometric point in this task is to recognize that the line of sight from the mountain top towards the horizon is tangent to the earth. We can then use a right triangle where one leg is tangent to a circle and the other leg is the radius of the circle to investigate this situation.

Inscribing a circle in a triangle II:

This problem solving task focuses on a remarkable fact which comes out of the construction of the inscribed circle in a triangle: the angle bisectors of the three angles of triangle ABC all meet in a point.

Text Resource

Why Tau Trumps Pi:

This informational text resource is intended to support reading in the content area. The author tries to convince the reader that two pi, or tau, occurs more often in mathematics than pi by itself. The author provides several examples and indicates the history behind society's choice of pi rather than tau.

Type: Text Resource

Student Resources

Vetted resources students can use to learn the concepts and skills in this topic.

The Lighthouse Problem:

This problem asks students to model phenomena on the surface of the earth by examining the visibility of the lamp in a lighthouse from a boat.

Solar Eclipse:

This problem solving task encourages students to explore why solar eclipses are rare by examining the radius of the sun and the furthest distance between the moon and the earth.

Eratosthenes and the circumference of the earth:

This problem solving task gives an interesting context for implementing ideas from geometry and trigonometry.

Triangles inscribed in a circle:

This problem solving task challenges students to use ideas about linear functions in order to determine when certain angles are right angles.

Inscribing a square in a circle:

This task provides an opportunity for students to apply triangle congruence theorems in an explicit, interesting context.

Two Wheels and a Belt:

This task combines two skills: making use of the relationship between a tangent segment to a circle and the radius touching that tangent segment, and computing lengths of circular arcs given the radii and central angles.

Right triangles inscribed in circles II:

Placing a Fire Hydrant:

This problem solving task asks students to place a fire hydrant so that it is equal distance from three given points.

Inscribing a triangle in a circle:

This problem introduces the circumcenter of a triangle and shows how it can be used to inscribe the triangle in a circle.

Circumcenter of a triangle:

This task shows that the three perpendicular bisectors of the sides of a triangle all meet in a point, using the characterization of the perpendicular bisector of a line segment as the set of points equidistant from the two ends of the segment.

Tangent Lines and the Radius of a Circle:

This problem solving task challenges students to find the perpendicular meeting point of a segment from the center of a circle and a tangent.

Setting Up Sprinklers:

This modeling task involves several different types of geometric knowledge and problem-solving: finding areas of sectors of circles, using trigonometric ratios to solve right triangles, and decomposing a complicated figure involving multiple circular arcs into parts whose areas can be found.

Neglecting the Curvature of the Earth:

This task applies geometric concepts, namely properties of tangents to circles and of right triangles, in a modeling situation. The key geometric point in this task is to recognize that the line of sight from the mountain top towards the horizon is tangent to the earth. We can then use a right triangle where one leg is tangent to a circle and the other leg is the radius of the circle to investigate this situation.

Inscribing a circle in a triangle II:

This problem solving task focuses on a remarkable fact which comes out of the construction of the inscribed circle in a triangle: the angle bisectors of the three angles of triangle ABC all meet in a point.

Parent Resources

Vetted resources caregivers can use to help students learn the concepts and skills in this topic.

The Lighthouse Problem:

This problem asks students to model phenomena on the surface of the earth by examining the visibility of the lamp in a lighthouse from a boat.

Solar Eclipse:

This problem solving task encourages students to explore why solar eclipses are rare by examining the radius of the sun and the furthest distance between the moon and the earth.

Eratosthenes and the circumference of the earth:

This problem solving task gives an interesting context for implementing ideas from geometry and trigonometry.

Triangles inscribed in a circle:

This problem solving task challenges students to use ideas about linear functions in order to determine when certain angles are right angles.

Inscribing a square in a circle:

This task provides an opportunity for students to apply triangle congruence theorems in an explicit, interesting context.

Two Wheels and a Belt:

This task combines two skills: making use of the relationship between a tangent segment to a circle and the radius touching that tangent segment, and computing lengths of circular arcs given the radii and central angles.

Right triangles inscribed in circles II:

Placing a Fire Hydrant:

This problem solving task asks students to place a fire hydrant so that it is equal distance from three given points.

Inscribing a triangle in a circle:

This problem introduces the circumcenter of a triangle and shows how it can be used to inscribe the triangle in a circle.

Circumcenter of a triangle:

This task shows that the three perpendicular bisectors of the sides of a triangle all meet in a point, using the characterization of the perpendicular bisector of a line segment as the set of points equidistant from the two ends of the segment.

Tangent Lines and the Radius of a Circle:

This problem solving task challenges students to find the perpendicular meeting point of a segment from the center of a circle and a tangent.

Setting Up Sprinklers:

This modeling task involves several different types of geometric knowledge and problem-solving: finding areas of sectors of circles, using trigonometric ratios to solve right triangles, and decomposing a complicated figure involving multiple circular arcs into parts whose areas can be found.