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Problems on Area and Circumference

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Videos, examples, lessons, and solutions to help Grade 7 students learn how to give an informal derivation of the relationship between the circumference and area of a circle.

New York State Common Core Math Grade 7, Module 3, Lesson 18

Lesson 18 Student Outcomes

• students learn how to examine the meaning of quarter circle and semicircle.
• Students solve area and perimeter problems for regions made out of rectangles, quarter circles, semicircles, and circles, including solving for unknown lengths when the area or perimeter is given.


• The area of a semicircular region is 1/2 of the area of a circle with the same radius.
• The area of a quarter of a circular region is 1/4 of the area of a circle with the same radius.
• If a problem asks you to use 22/7 for π, look for ways to use fraction arithmetic to simplify your computations in the problem.
• Problems that involve the composition of several shapes may be decomposed in more than one way.

Lesson 18 Opening Exercise

Draw a circle of diameter 12 cm and a square of side length 12 cm on grid paper. Determine the area of the square and the circle.
Brainstorm some methods for finding half the area of the square and half the area of the circle.
Find the area of half of the square and half of the circle, and explain to a partner how you arrived at the area.
What is the ratio of the new area to the original area for the square and for the circle?
Find the area of one-fourth of the square and the circle, first by folding and then by another method. What is the ratio of the new area to the original area for the square and for the circle?
Write an algebraic expression that will express the area of a semicircle and the area of a quarter circle.

Example 1

Find the area of the following semicircle.
What is the area of the quarter circle?

Example 2

Marjorie is designing a new set of placemats for her dining room table. She sketched a drawing of the placement on graph paper. The diagram represents the area of the placemat consisting of a rectangle and two semicircles at either end. Each square on the grid measures 4 inches in length.
Find the area of the entire placemat. Explain your thinking regarding the solution to this problem.
If Marjorie wants to make six placemats, how many square inches of fabric will she need?
Marjorie decides that she wants to sew on a contrasting band of material around the edge of the placemats. How much binding material will Marjorie need?

Example 3

The circumference of a circle is 24πcm. What is the exact area of the circle?
Draw a diagram to assist you in solving the problem.
What information is needed to solve the problem? Next, find the area.


1. Find the area of a circle with a diameter of 42 cm. Use π = 22/7.

2. The circumference of a circle is 9π cm.
a. What is the diameter?
b. What is the radius?
c. What is the area?

3. If students only know the radius of a circle, what other measures could they determine? Explain how students would use the radius to find the other parts.

4. Find the area in the rectangle between the two quarter circles if AB = 7 ft., FB = 9 ft., and HD = 7 ft. Use π = 22/7.

Lesson 18 Exit Ticket Sample Solutions

1. Ken’s landscape gardening business creates odd shaped lawns which include semicircles. Find the area of this semicircular section of the lawn in this design. Use 22/7 for π.

2. In the figure below, Ken’s company has placed sprinkler heads at the center of the two small semicircles. The radius of the sprinklers is 5ft. If the area in the larger semicircular area is the shape of the entire lawn, how much of the lawn will not be watered? Give your answer in terms of π and to the nearest tenth. Explain your thinking.

Rotate to landscape screen format on a mobile phone or small tablet to use the Mathway widget, a free math problem solver that answers your questions with step-by-step explanations.

You can use the free Mathway calculator and problem solver below to practice Algebra or other math topics. Try the given examples, or type in your own problem and check your answer with the step-by-step explanations.

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