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Prealgebra 2e

9.5 Solve Geometry Applications: Circles and Irregular Figures

Prealgebra 2e9.5 Solve Geometry Applications: Circles and Irregular Figures
  1. Preface
  2. 1 Whole Numbers
    1. Introduction
    2. 1.1 Introduction to Whole Numbers
    3. 1.2 Add Whole Numbers
    4. 1.3 Subtract Whole Numbers
    5. 1.4 Multiply Whole Numbers
    6. 1.5 Divide Whole Numbers
    7. Key Terms
    8. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  3. 2 The Language of Algebra
    1. Introduction to the Language of Algebra
    2. 2.1 Use the Language of Algebra
    3. 2.2 Evaluate, Simplify, and Translate Expressions
    4. 2.3 Solving Equations Using the Subtraction and Addition Properties of Equality
    5. 2.4 Find Multiples and Factors
    6. 2.5 Prime Factorization and the Least Common Multiple
    7. Key Terms
    8. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  4. 3 Integers
    1. Introduction to Integers
    2. 3.1 Introduction to Integers
    3. 3.2 Add Integers
    4. 3.3 Subtract Integers
    5. 3.4 Multiply and Divide Integers
    6. 3.5 Solve Equations Using Integers; The Division Property of Equality
    7. Key Terms
    8. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  5. 4 Fractions
    1. Introduction to Fractions
    2. 4.1 Visualize Fractions
    3. 4.2 Multiply and Divide Fractions
    4. 4.3 Multiply and Divide Mixed Numbers and Complex Fractions
    5. 4.4 Add and Subtract Fractions with Common Denominators
    6. 4.5 Add and Subtract Fractions with Different Denominators
    7. 4.6 Add and Subtract Mixed Numbers
    8. 4.7 Solve Equations with Fractions
    9. Key Terms
    10. Key Concepts
    11. Exercises
      1. Review Exercises
      2. Practice Test
  6. 5 Decimals
    1. Introduction to Decimals
    2. 5.1 Decimals
    3. 5.2 Decimal Operations
    4. 5.3 Decimals and Fractions
    5. 5.4 Solve Equations with Decimals
    6. 5.5 Averages and Probability
    7. 5.6 Ratios and Rate
    8. 5.7 Simplify and Use Square Roots
    9. Key Terms
    10. Key Concepts
    11. Exercises
      1. Review Exercises
      2. Practice Test
  7. 6 Percents
    1. Introduction to Percents
    2. 6.1 Understand Percent
    3. 6.2 Solve General Applications of Percent
    4. 6.3 Solve Sales Tax, Commission, and Discount Applications
    5. 6.4 Solve Simple Interest Applications
    6. 6.5 Solve Proportions and their Applications
    7. Key Terms
    8. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  8. 7 The Properties of Real Numbers
    1. Introduction to the Properties of Real Numbers
    2. 7.1 Rational and Irrational Numbers
    3. 7.2 Commutative and Associative Properties
    4. 7.3 Distributive Property
    5. 7.4 Properties of Identity, Inverses, and Zero
    6. 7.5 Systems of Measurement
    7. Key Terms
    8. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  9. 8 Solving Linear Equations
    1. Introduction to Solving Linear Equations
    2. 8.1 Solve Equations Using the Subtraction and Addition Properties of Equality
    3. 8.2 Solve Equations Using the Division and Multiplication Properties of Equality
    4. 8.3 Solve Equations with Variables and Constants on Both Sides
    5. 8.4 Solve Equations with Fraction or Decimal Coefficients
    6. Key Terms
    7. Key Concepts
    8. Exercises
      1. Review Exercises
      2. Practice Test
  10. 9 Math Models and Geometry
    1. Introduction
    2. 9.1 Use a Problem Solving Strategy
    3. 9.2 Solve Money Applications
    4. 9.3 Use Properties of Angles, Triangles, and the Pythagorean Theorem
    5. 9.4 Use Properties of Rectangles, Triangles, and Trapezoids
    6. 9.5 Solve Geometry Applications: Circles and Irregular Figures
    7. 9.6 Solve Geometry Applications: Volume and Surface Area
    8. 9.7 Solve a Formula for a Specific Variable
    9. Key Terms
    10. Key Concepts
    11. Exercises
      1. Review Exercises
      2. Practice Test
  11. 10 Polynomials
    1. Introduction to Polynomials
    2. 10.1 Add and Subtract Polynomials
    3. 10.2 Use Multiplication Properties of Exponents
    4. 10.3 Multiply Polynomials
    5. 10.4 Divide Monomials
    6. 10.5 Integer Exponents and Scientific Notation
    7. 10.6 Introduction to Factoring Polynomials
    8. Key Terms
    9. Key Concepts
    10. Exercises
      1. Review Exercises
      2. Practice Test
  12. 11 Graphs
    1. Graphs
    2. 11.1 Use the Rectangular Coordinate System
    3. 11.2 Graphing Linear Equations
    4. 11.3 Graphing with Intercepts
    5. 11.4 Understand Slope of a Line
    6. Key Terms
    7. Key Concepts
    8. Exercises
      1. Review Exercises
      2. Practice Test
  13. A | Cumulative Review
  14. B | Powers and Roots Tables
  15. C | Geometric Formulas
  16. Answer Key
    1. Chapter 1
    2. Chapter 2
    3. Chapter 3
    4. Chapter 4
    5. Chapter 5
    6. Chapter 6
    7. Chapter 7
    8. Chapter 8
    9. Chapter 9
    10. Chapter 10
    11. Chapter 11
  17. Index

Learning Objectives

By the end of this section, you will be able to:

  • Use the properties of circles
  • Find the area of irregular figures
Be Prepared 9.13

Before you get started, take this readiness quiz.

Evaluate x2x2 when x=5.x=5.
If you missed this problem, review Example 2.15.

Be Prepared 9.14

Using 3.143.14 for π,π, approximate the (a) circumference and (b) the area of a circle with radius 88 inches.
If you missed this problem, review Example 5.39.

Be Prepared 9.15

Simplify 227(0.25)2227(0.25)2 and round to the nearest thousandth.
If you missed this problem, review Example 5.36.

In this section, we’ll continue working with geometry applications. We will add several new formulas to our collection of formulas. To help you as you do the examples and exercises in this section, we will show the Problem Solving Strategy for Geometry Applications here.

Problem Solving Strategy for Geometry Applications

  1. Step 1. Read the problem and make sure you understand all the words and ideas. Draw the figure and label it with the given information.
  2. Step 2. Identify what you are looking for.
  3. Step 3. Name what you are looking for. Choose a variable to represent that quantity.
  4. Step 4. Translate into an equation by writing the appropriate formula or model for the situation. Substitute in the given information.
  5. Step 5. Solve the equation using good algebra techniques.
  6. Step 6. Check the answer in the problem and make sure it makes sense.
  7. Step 7. Answer the question with a complete sentence.

Use the Properties of Circles

Do you remember the properties of circles from Decimals and Fractions Together? We’ll show them here again to refer to as we use them to solve applications.

Properties of Circles

An image of a circle is shown. There is a line drawn through the widest part at the center of the circle with a red dot indicating the center of the circle. The line is labeled d. The two segments from the center of the circle to the outside of the circle are each labeled r.
  • rr is the length of the radius
  • dd is the length of the diameter
  • d=2rd=2r
  • Circumference is the perimeter of a circle. The formula for circumference is
    C=2πrC=2πr
  • The formula for area of a circle is
    A=πr2A=πr2

Remember, that we approximate ππ with 3.143.14 or 227227 depending on whether the radius of the circle is given as a decimal or a fraction. If you use the ππ key on your calculator to do the calculations in this section, your answers will be slightly different from the answers shown. That is because the ππ key uses more than two decimal places.

Example 9.41

A circular sandbox has a radius of 2.52.5 feet. Find the circumference and area of the sandbox.

Try It 9.81


A circular mirror has radius of 55 inches. Find the circumference and area of the mirror.

Try It 9.82

A circular spa has radius of 4.54.5 feet. Find the circumference and area of the spa.

We usually see the formula for circumference in terms of the radius rr of the circle:

C=2πrC=2πr

But since the diameter of a circle is two times the radius, we could write the formula for the circumference in terms ofd.ofd.

C=2πrUsing the commutative property, we getC=π·2rThen substitutingd=2rC=π·dSoC=πdC=2πrUsing the commutative property, we getC=π·2rThen substitutingd=2rC=π·dSoC=πd

We will use this form of the circumference when we’re given the length of the diameter instead of the radius.

Example 9.42

A circular table has a diameter of four feet. What is the circumference of the table?

Try It 9.83

Find the circumference of a circular fire pit whose diameter is 5.55.5 feet.

Try It 9.84

If the diameter of a circular trampoline is 1212 feet, what is its circumference?

Example 9.43

Find the diameter of a circle with a circumference of 47.147.1 centimeters.

Try It 9.85

Find the diameter of a circle with circumference of 94.294.2 centimeters.

Try It 9.86

Find the diameter of a circle with circumference of 345.4345.4 feet.

Find the Area of Irregular Figures

So far, we have found area for rectangles, triangles, trapezoids, and circles. An irregular figure is a figure that is not a standard geometric shape. Its area cannot be calculated using any of the standard area formulas. But some irregular figures are made up of two or more standard geometric shapes. To find the area of one of these irregular figures, we can split it into figures whose formulas we know and then add the areas of the figures.

Example 9.44

Find the area of the shaded region.

An image of an attached horizontal rectangle and a vertical rectangle is shown. The top is labeled 12, the side of the horizontal rectangle is labeled 4. The side is labeled 10, the width of the vertical rectangle is labeled 2.
Try It 9.87

Find the area of each shaded region:

A blue geometric shape is shown. It looks like a horizontal rectangle attached to a vertical rectangle. The top is labeled as 8, the width of the horizontal rectangle is labeled as 2. The side is labeled as 6, the width of the vertical rectangle is labeled as 3.
Try It 9.88

Find the area of each shaded region:

A blue geometric shape is shown. It looks like a horizontal rectangle attached to a vertical rectangle. The top is labeled as 14, the width of the horizontal rectangle is labeled as 5. The side is labeled as 10, the width of the missing space is labeled as 6.

Example 9.45

Find the area of the shaded region.

A blue geometric shape is shown. It looks like a rectangle with a triangle attached to the top on the right side. The left side is labeled 4, the top 5, the bottom 8, the right side 7.
Try It 9.89

Find the area of each shaded region.

A blue geometric shape is shown. It looks like a rectangle with a triangle attached to the lower right side. The base of the rectangle is labeled 8, the height of the rectangle is labeled 4. The distance from the top of the rectangle to where the triangle begins is labeled 3, the top of the triangle is labeled 3.
Try It 9.90

Find the area of each shaded region.

A blue geometric shape is shown. It looks like a rectangle with an equilateral triangle attached to the top. The base of the rectangle is labeled 12, each side is labeled 5. The base of the triangle is split into two pieces, each labeled 2.5.

Example 9.46

A high school track is shaped like a rectangle with a semi-circle (half a circle) on each end. The rectangle has length 105105 meters and width 6868 meters. Find the area enclosed by the track. Round your answer to the nearest hundredth.

A track is shown, shaped like a rectangle with a semi-circle attached to each side.
Try It 9.91

Find the area:

A shape is shown. It is a blue rectangle with a portion of the rectangle missing. There is a red circle the same height as the rectangle attached to the missing side of the rectangle. The top of the rectangle is labeled 15, the height is labeled 9.
Try It 9.92

Find the area:

A blue geometric shape is shown. It appears to be two trapezoids with a semicircle at the top. The base of the semicircle is labeled 5.2. The height of the trapezoids is labeled 6.5. The combined base of the trapezoids is labeled 3.3.

Section 9.5 Exercises

Practice Makes Perfect

Use the Properties of Circles

In the following exercises, solve using the properties of circles.

217.

The lid of a paint bucket is a circle with radius 77 inches. Find the circumference and area of the lid.

218.

An extra-large pizza is a circle with radius 88 inches. Find the circumference and area of the pizza.

219.

A farm sprinkler spreads water in a circle with radius of 8.58.5 feet. Find the circumference and area of the watered circle.

220.

A circular rug has radius of 3.53.5 feet. Find the circumference and area of the rug.

221.

A reflecting pool is in the shape of a circle with diameter of 2020 feet. What is the circumference of the pool?

222.

A turntable is a circle with diameter of 1010 inches. What is the circumference of the turntable?

223.

A circular saw has a diameter of 1212 inches. What is the circumference of the saw?

224.

A round coin has a diameter of 33 centimeters. What is the circumference of the coin?

225.

A barbecue grill is a circle with a diameter of 2.22.2 feet. What is the circumference of the grill?

226.

The top of a pie tin is a circle with a diameter of 9.59.5 inches. What is the circumference of the top?

227.

A circle has a circumference of 163.28163.28 inches. Find the diameter.

228.

A circle has a circumference of 59.6659.66 feet. Find the diameter.

229.

A circle has a circumference of 17.2717.27 meters. Find the diameter.

230.

A circle has a circumference of 80.0780.07 centimeters. Find the diameter.

In the following exercises, find the radius of the circle with given circumference.

231.

A circle has a circumference of 150.72150.72 feet.

232.

A circle has a circumference of 251.2251.2 centimeters.

233.

A circle has a circumference of 40.8240.82 miles.

234.

A circle has a circumference of 78.578.5 inches.

Find the Area of Irregular Figures

In the following exercises, find the area of the irregular figure. Round your answers to the nearest hundredth.

235.
A geometric shape is shown. It is a horizontal rectangle attached to a vertical rectangle. The top is labeled 6, the height of the horizontal rectangle is labeled 2, the distance from the edge of the horizontal rectangle to the start of the vertical rectangle is 4, the base of the vertical rectangle is 2, the right side of the shape is 4.
236.
A geometric shape is shown. It is an L-shape. The base is labeled 10, the right side 1, the top and left side are each labeled 4.
237.
A geometric shape is shown. It is a sideways U-shape. The top is labeled 6, the left side is labeled 6. An inside horizontal piece is labeled 3. Each of the vertical pieces on the right are labeled 2.
238.
A geometric shape is shown. It is a U-shape. The base is labeled 7. The right side is labeled 5. The two horizontal lines at the top and the vertical line on the inside are all labeled 3.
239.
A geometric shape is shown. It is a rectangle with a triangle attached to the bottom left side. The top is labeled 4. The right side is labeled 10. The base is labeled 9. The vertical line from the top of the triangle to the top of the rectangle is labeled 3.
240.
A trapezoid is shown. The bases are labeled 5 and 10, the height is 5.
241.
Two triangles are shown. They appear to be right triangles. The bases are labeled 3, the heights 4, and the longest sides 5.
242.
A geometric shape is shown. It appears to be composed of two triangles. The shared base of both triangles is 8, the heights are both labeled 6.
243.
A geometric shape is shown. It is composed of two trapezoids. The base is labeled 10. The height of one trapezoid is 2. The horizontal and vertical sides are all labeled 5.
244.
A geometric shape is shown. It is a trapezoid attached to a triangle. The base of the triangle is labeled 6, the height is labeled 5. The height of the trapezoid is 6, one base is 3.
245.
A geometric shape is shown. It is a rectangle with a triangle and another rectangle attached. The left side is labeled 8, the bottom is 8, the right side is 13, and the width of the smaller rectangle is 2.
246.
A geometric shape is shown. It is a rectangle with a triangle and another rectangle attached. The left side is labeled 12, the right side 7, the base 6. The width of the smaller rectangle is labeled 1.
247.
A geometric shape is shown. It is a rectangle attached to a semi-circle. The base of the rectangle is labeled 5, the height is 7.
248.
A geometric shape is shown. It is a rectangle attached to a semi-circle. The base of the rectangle is labeled 10, the height is 6. The portion of the rectangle on the left of the semi-circle is labeled 5, the portion on the right is labeled 2.
249.
A geometric shape is shown. A triangle is attached to a semi-circle. The base of the triangle is labeled 4. The height of the triangle and the diameter of the circle are 8.
250.
A geometric shape is shown. A triangle is attached to a semi-circle. The height of the triangle is labeled 4. The base of the triangle, also the diameter of the semi-circle, is labeled 4.
251.
A geometric shape is shown. It is a rectangle attached to a semi-circle. The base of the rectangle is labeled 5, the height is 7.
252.
A geometric shape is shown. A trapezoid is shown with a semi-circle attached to the top. The diameter of the circle, which is also the top of the trapezoid, is labeled 8. The height of the trapezoid is 6. The bottom of the trapezoid is 13.
253.
A geometric shape is shown. It is a rectangle with a triangle attached to the top on the left side and a circle attached to the top right corner. The diameter of the circle is labeled 5. The height of the triangle is labeled 5, the base is labeled 4. The height of the rectangle is labeled 6, the base 11.
254.
A geometric shape is shown. It is a trapezoid with a triangle attached to the top, and a circle attached to the triangle. The diameter of the circle is 4. The height of the triangle is 5, the base of the triangle, which is also the top of the trapezoid, is 6. The bottom of the trapezoid is 9. The height of the trapezoid is 7.

In the following exercises, solve.

255.

A city park covers one block plus parts of four more blocks, as shown. The block is a square with sides 250250 feet long, and the triangles are isosceles right triangles. Find the area of the park.

A square is shown with four triangles coming off each side.
256.

A gift box will be made from a rectangular piece of cardboard measuring 1212 inches by 2020 inches, with squares cut out of the corners of the sides, as shown. The sides of the squares are 33 inches. Find the area of the cardboard after the corners are cut out.

A rectangle is shown. Each corner has a gray shaded square. There are dotted lines drawn across the side of each square attached to the next square.
257.

Perry needs to put in a new lawn. His lot is a rectangle with a length of 120120 feet and a width of 100100 feet. The house is rectangular and measures 5050 feet by 4040 feet. His driveway is rectangular and measures 2020 feet by 3030 feet, as shown. Find the area of Perry’s lawn.

A rectangular lot is shown. In it is a home shaped like a rectangle attached to a rectangular driveway.
258.

Denise is planning to put a deck in her back yard. The deck will be a 20-ft20-ft by 12-ft12-ft rectangle with a semicircle of diameter 66 feet, as shown below. Find the area of the deck.

A picture of a deck is shown. It is shaped like a rectangle with a semi-circle attached to the top on the left side.

Everyday Math

259.

Area of a Tabletop Yuki bought a drop-leaf kitchen table. The rectangular part of the table is a 1-ft1-ft by 3-ft3-ft rectangle with a semicircle at each end, as shown. Find the area of the table with one leaf up. Find the area of the table with both leaves up.

An image of a table is shown. There is a rectangular portion attached to a semi-circular portion. There is another semi-circular leaf folded down on the other side of the rectangle.
260.

Painting Leora wants to paint the nursery in her house. The nursery is an 8-ft8-ft by 10-ft10-ft rectangle, and the ceiling is 88 feet tall. There is a 3-ft3-ft by 6.5-ft6.5-ft door on one wall, a 3-ft3-ft by 6.5-ft6.5-ft closet door on another wall, and one 4-ft4-ft by 3.5-ft3.5-ft window on the third wall. The fourth wall has no doors or windows. If she will only paint the four walls, and not the ceiling or doors, how many square feet will she need to paint?

Writing Exercises

261.

Describe two different ways to find the area of this figure, and then show your work to make sure both ways give the same area.

A geometric shape is shown. It is a vertical rectangle attached to a horizontal rectangle. The width of the vertical rectangle is 3, the left side is labeled 6, the bottom is labeled 9, and the width of the horizontal rectangle is labeled 3. The top of the horizontal rectangle is labeled 6, and the distance from the top of that rectangle to the top of the other rectangle is labeled 3.
262.

A circle has a diameter of 1414 feet. Find the area of the circle using 3.143.14 forππ using 227227 for π.π. Which calculation to do prefer? Why?

Self Check

After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

.

After looking at the checklist, do you think you are well prepared for the next section? Why or why not?

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