Geometry

Lesson 6

Math

Unit 6

7th Grade

Lesson 6 of 21

Objective


Determine the relationship between the circumference and diameter of a circle and use it to solve problems. 

Common Core Standards


Core Standards

  • 7.G.B.4 — Know the formulas for the area and circumference of a circle and use them to solve problems; give an informal derivation of the relationship between the circumference and area of a circle.

Foundational Standards

  • 7.RP.A.2

Criteria for Success


  1. Understand there is a proportional relationship between the circumference of any circle and its diameter.
  2. Determine that the ratio of the circumference of a circle to its diameter is equivalent to $${\pi}$$.
  3. Know the formula that relates the circumference and diameter of a circle: $$C={\pi} d$$.
  4. Use the formula $$C={\pi} d$$  to solve problems. 

Tips for Teachers


  • Lessons 6 and 7 focus on the relationship between a circle’s circumference and its diameter. In Lesson 6, students determine this relationship and the formula that represents it, making connections to proportional relationships and constant of proportionality. In Lesson 7, students will solve real-world and mathematical problems involving circumference, including problems involving semi-circles. 
  • Consider having students create their own reference sheet to add to and refer back to throughout the unit as needed. This can be placed in a sheet protector or created on cardstock. Alternatively, once both formulas for circles have been determined, students may be given a state-provided or pre-created reference sheet to refer to as needed. Here is the Massachusetts Comprehensive Assessment System Grade 7 Mathematics Reference Sheet.

Lesson Materials

  • Ruler (1 per student)
  • Graph Paper (2-3 sheets per student)
  • String (1 per student)
  • Circles handout (1 per student)
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Anchor Problems


Problem 1

Using the circles handout, measure the circumference and diameter of each circle and record your results in the table below. 

Circle Diameter ($$d$$)

Circumference ($$C$$)

Ratio of $$C/d$$
A      
B      
C      
D      
E      
F      
G      

a.   Create a graph to show the relationship between circumference and diameter. Place the values for diameter along the $$x$$-axis and the values for circumference along the $$y$$-axis. Plot the points from the table. and then sketch a line through the points that best shows the relationship. (Your line will not pass through all of the points exactly.) 

b.   What is the constant of proportionality? Write an equation to relate the circumference and diameter of any circle. 

Guiding Questions

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Problem 2

Use the relationship between circumference and diameter to solve the two problems below.

a.   The diameter of a circular clock is 12 inches. What is the circumference of the clock?

b.   The distance around a circular cookie is approximately 18.84 cm. What is the diameter of the cookie?

Guiding Questions

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Problem Set

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Target Task


Problem 1

Describe the relationship between the circumference of a circle and its diameter. 

Problem 2

The top of a can of tuna is in the shape of a circle. If the distance around the top is approximately 251.2 mm, what is the diameter of the top of the can of tuna? What is the radius of the top of the can of tuna?

Student Response

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Additional Practice


The following resources include problems and activities aligned to the objective of the lesson that can be used for additional practice or to create your own problem set.

  • Include problems where students are given either the circumference or the diameter and asked to find the other value.
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Lesson 5

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Lesson 7

Lesson Map

A7CB09C2-D12F-4F55-80DB-37298FF0A765

Topic A: Angle Relationships

Topic B: Circles

Topic C: Building Polygons and Triangles

Topic D: Solid Figures

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