In previous lessons or grades, you may have encountered a four-step problem solving process. Click on each step below.

In this section, you will step through the problem-solving process to solve a problem involving the area of a composite figure. Then, you will practice using the problem-solving process on your own.

Problem

Six circles are used to create an art deco design. The diameter of each circle is 8.5 centimeters, and they are arranged so that they fit exactly inside one rectangle, as shown in the figure below.

What is the approximate area of the green, or shaded region, in the art deco design?

To solve this problem, use the four-step problem-solving model.

Step 1: Read, understand, and interpret the problem.

  • What information is presented?

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    The diameter of each circle is 8.5 centimeters.
    The six circles create a rectangle that has a width of three circles and a length of two circles. Close Pop Up
  • What is the problem asking you to find?

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    You are looking for the area of the shaded region, which is inside the rectangle, but outside the circles. Close Pop Up
  • What information do you not need?

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    There does not appear to be any extra information in the problem. Close Pop Up
Step 2: Make a plan.

  • Draw a picture.

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    Possible picture:
    Close Pop Up
Step 3: Implement your plan.

  • What formulas do you need?

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    Area of a circle: A = πr2

    Area of a rectangle: A = lw or A = bh Close Pop Up
  • What information can you interpret from the diagram, table, or other given information?

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    The length of the rectangle is equal to the combined diameters of two circles. The width of the rectangle is equal to the combined diameters of three circles. To calculate the area of each circle, you need to know the radius, which is half the diameter. Close Pop Up
  • Solve the problem.

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    Area of Shaded Region = Area of Rectangle − 6 × Area of Circle
    Area of Shaded Region = (2 × 8.5 cm)(3 × 8.5 cm) − 6(π)(4.25 cm)2
    Area of Shaded Region = (17 cm)(25.5 cm) − 6(π)(18.0625 cm2)
    Area of Shaded Region ≈ 433.5 cm22− 340.30 cm2
    Area of Shaded Region ≈ 93.2 cm2 Close Pop Up
Step 4: Evaluate your answer.

  • Does your answer make sense?

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    Yes. The area of one rectangle is about 400 square centimeters, and the area of one circle is about 50 square centimeters. 400 − 6(50) = 100, which is close to the calculated area. Close Pop Up
  • Did you answer the question that was asked?

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    Yes. The question asked for the area of the shaded region, the area of the rectangle not in the circles. Close Pop Up
  • Are your units correct?

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    Yes. Area always has square units. Since the original dimensions were in centimeters, the answer should have units of square centimeters.Close Pop Up

Pause and Reflect

Why is it important to evaluate your answer?

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If you evaluate your answer, then you can have more confidence that your answer is correct.Close Pop Up

What are some ways to evaluate your answer?

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You can evaluate your answer by rounding the dimensions to estimate your answer. You can also work backwards to make sure that your calculations were performed correctly.Close Pop Up

Practice

  1. The floor plan for a family room is shown in the figure below.

    What is the approximate area of the floor of the living room?

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    Rectangles have four right angles and opposite sides that are congruent. Look for ways to partition the figure around right angles.Close Pop Up

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    Check Your Answer


    Area of Rectangle:
    A = bh = (12 ft)(8 ft) = 96 ft2

    Area of Quarter Circle:
    Calculate the area of the circle, and then divide the area of the circle by 4 to find the area of a quarter circle.
    A = πr2 ≈ (3.14)(3 ft)2 ≈28.26 ft2
    A ≈28.26/4 ≈ 7.065 ft2

    Area of Trapezoid:
    A = 1 over 2 1 2 (b1 + b2)h = 1 over 2 1 2 (20 ft + 12 ft)(8 ft) A = 128 ft2

    Total Area = Area of Rectangle + Area of Quarter Circle + Area of Trapezoid

    Total Area = 96 ft2 + 7.065 ft2 + 128 ft2
    Total Area = 231.065 ft2 Close Pop Up
  2. The diagram below shows a pasture that is shaped like a parallelogram. Inside the pasture is a rectangular shed and a trough that is shaped like a rectangle with two semicircles at each end. What is the approximate area of the shaded region, representing the amount of grass available for grazing? Round your answer to the nearest tenth.

    Interactive exercise. Assistance may be required. Use the grid below to record your answer. Type your answer in the boxes in front of and behind the decimal. Click inside each box to enter the numeral that belongs in the box. Click the bubble beneath the numeral to shade the bubble that matches the numeral. Incorrect portions of your answer will be shaded gray.

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    What is the area of the rectangular shed and the trough made up of two semicircles and one rectangle? What area formulas do you need to use in order to solve this problem?Close Pop Up
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    Possible Solution:
    Area of Grazing Land = Area of Pasture − Area of Shed − Area of Trough
    Area of Grazing Land = Area of Parallelogram − Area of Rectangle − (Area of Semicircle + Area of Rectangle + Area of Semicircle)

    Area of Parallelogram:
    A = bh = (25 yd)(8 yd + 3 yd) = (25 yd)(11 yd) = 275 yd2

    Area of Rectangle (Shed):
    A = bh = (8.5 yd)(3 yd) = 25.5 yd2

    Area of Trough:
    A = 1 over 2 1 2 πr2 + bh + 1 over 2 1 2 πr2
    A1 over 2 1 2 (3.14)(1 yd)2 + (2.8 yd)(2 yd) + 1 over 2 1 2 (3.14)(1 yd)2
    A ≈ 1.57 yd2 + 5.6 yd2 + 1.57 yd2
    A ≈ 8.7 yd2

    Area of Grazing Land = 275 yd2 − 25.5 yd2 − 8.7 yd2 = 240.8 yd2 Close Pop Up