Area Problems with Circular Regions

Examples, videos, and solutions to help Grade 7 students learn how to determine the area of composite figures and of missing regions using composition and decomposition of polygons.

New York State Common Core Math Grade 7, Module 6, Lesson 22

Worksheets for Grade 7

Lesson 22 Student Outcomes

• Students determine the area of composite figures and of missing regions using composition and decomposition of polygons.

Lesson 22 Summary

• In calculating composite figures with circular regions, it is important to identify relevant geometric areas; for example, identify relevant rectangles or squares that are part of a figure with a circular region. • Next, determine which areas should be subtracted or added based on their positions in the diagram. • Be sure to note whether a question asks for the exact or approximate area.

Lesson 22 Classwork Example 1
a. The circle to the right has a diameter of 12 cm. Calculate the area of the shaded region.
b. Sasha, Barry, and Kyra wrote three different expressions for the area of the shaded region. Describe what each student was thinking about the problem based on their expression.

Exercise 1
a. Find the area of the shaded region of the circle to the right.
b. Explain how the expression you used represents the area of the shaded region.

Exercise 2
Calculate the area of the figure below that consists of a rectangle and two quarter circles, each with the same radius. Leave your answer in terms of pi.

Example 2
The square in this figure has a side length of 14 inches. The radius of the quarter circle is 7 inches.
a. Estimate the shaded area.
b. What is the exact area of the shaded region?
c. What is the approximate area using π = 22/7?

Exercise 3
The vertices A and B of rectangle ABCD are centers of circles each with a radius of inches.
a. Find the exact area of the shaded region.
b. Find the approximate area using π = 22/7.
c. Find the area to the nearest hundredth using your key on your calculator.

Exercise 4
The diameter of the circle is 12 in. Write and explain a numerical expression that represents the area.

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