9.5 Solve Geometry Applications: Circles and Irregular Figures

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

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

Remember, that we approximate $π$ with $3.14$ or $\frac{22}{7}$ depending on whether the radius of the circle is given as a decimal or a fraction. If you use the $π$ 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 $π$ key uses more than two decimal places.

Example 1

A circular sandbox has a radius of $2.5$ feet. Find the ⓐ circumference and ⓑ area of the sandbox.

ⓐ Step 1. Read the problem. Draw the figure and label it with the given information.
Step 2. Identify what you are looking for.	the circumference of the circle
Step 3. Name. Choose a variable to represent it.	Let c = circumference of the circle
Step 4. Translate. Write the appropriate formula Substitute	$C equals 2 π r$ $C equals 2 π open parenthesis 2.5 close parenthesis$
Step 5. Solve the equation.	$C approximately equals 2 open parenthesis 3.14 close parenthesis open parenthesis 2.5 close parenthesis$ $C approximately equals 15 ft$
Step 6. Check. Does this answer make sense? Yes. If we draw a square around the circle, its sides would be 5 ft (twice the radius), so its perimeter would be 20 ft. This is slightly more than the circle's circumference, 15.7 ft.
Step 7. Answer the question.	The circumference of the sandbox is 15.7 feet.

ⓑ Step 1. Read the problem. Draw the figure and label it with the given information.
Step 2. Identify what you are looking for.	the area of the circle
Step 3. Name. Choose a variable to represent it.	Let A = the area of the circle
Step 4. Translate. Write the appropriate formula Substitute	$A = π r^{2}$ $A equals π open parenthesis 2.5 close parenthesis squared$
Step 5. Solve the equation.	$A approximately equals open parenthesis 3.14 close parenthesis open parenthesis 2.5 close parenthesis squared$ $A approximately equals 19.625 sq. ft$
Step 6. Check. Yes. If we draw a square around the circle, its sides would be 5 ft, as shown in part ⓐ. So the area of the square would be 25 sq. ft. This is slightly more than the circle's area, 19.625 sq. ft.
Step 7. Answer the question.	The area of the circle is 19.625 square feet.

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

$C equals 2 π r$

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

$\begin{matrix} C = 2 π r \\ Using the commutative property, we get & C = π \cdot 2 r \\ Then substituting d = 2 r & C = π \cdot d \\ So & C = π d \end{matrix}$

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

Example 2

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

Step 1. Read the problem. Draw the figure and label it with the given information.
Step 2. Identify what you are looking for.	the circumference of the table
Step 3. Name. Choose a variable to represent it.	Let c = the circumference of the table
Step 4. Translate. Write the appropriate formula for the situation. Substitute.	$C = π d$ $C equals π open parenthesis 4 close parenthesis$
Step 5. Solve the equation, using 3.14 for $π .$	$C approximately equals open parenthesis 3.14 close parenthesis open parenthesis 4 close parenthesis$ $C approximately equals 12.56 feet$
Step 6. Check: If we put a square around the circle, its side would be 4. The perimeter would be 16. It makes sense that the circumference of the circle, 12.56, is a little less than 16.
Step 7. Answer the question.	The diameter of the table is 12.56 feet.

Example 3

Find the diameter of a circle with a circumference of $47.1$ centimeters.

Step 1. Read the problem. Draw the figure and label it with the given information.
Step 2. Identify what you are looking for.	the diameter of the circle
Step 3. Name. Choose a variable to represent it.	Let d = the diameter of the circle
Step 4. Translate.
Write the formula. Substitute, using 3.14 to approximate $π$ .
Step 5. Solve.	$A mathematical expression shows the fraction 47.1 over 3.14 approximately equal to the fraction 3.14d over 3.14.$
Step 6. Check: $47.1 equals ? open parenthesis 3.14 close parenthesis open parenthesis 15 close parenthesis$ $47.1 equals 47.1 ✓$
Step 7. Answer the question.	The diameter of the circle is approximately 15 centimeters.

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.

Key Concepts

Problem Solving Strategy for Geometry Applications
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. Identify what you are looking for.
3. Name what you are looking for. Choose a variable to represent that quantity.
4. Translate into an equation by writing the appropriate formula or model for the situation. Substitute in the given information.
5. Solve the equation using good algebra techniques.
6. Check the answer in the problem and make sure it makes sense.
7. Answer the question with a complete sentence.
Properties of Circles
- $d equals 2 r$
- Circumference: $C equals 2 π r$ or $C = π d$
- Area: $A = π r^{2}$

Practice Makes Perfect

Use the Properties of Circles

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

The lid of a paint bucket is a circle with radius $7$ inches. Find the ⓐ circumference and ⓑ area of the lid.

ⓐ 43.96 in.
ⓑ 153.86 sq. in.

An extra-large pizza is a circle with radius $8$ inches. Find the ⓐ circumference and ⓑ area of the pizza.

A farm sprinkler spreads water in a circle with radius of $8.5$ feet. Find the ⓐ circumference and ⓑ area of the watered circle.

ⓐ 53.38 ft
ⓑ 226.865 sq. ft

A circular rug has radius of $3.5$ feet. Find the ⓐ circumference and ⓑ area of the rug.

A reflecting pool is in the shape of a circle with diameter of $20$ feet. What is the circumference of the pool?

62.8 ft

A turntable is a circle with diameter of $10$ inches. What is the circumference of the turntable?

A circular saw has a diameter of $12$ inches. What is the circumference of the saw?

37.68 in.

A round coin has a diameter of $3$ centimeters. What is the circumference of the coin?

A barbecue grill is a circle with a diameter of $2.2$ feet. What is the circumference of the grill?

6.908 ft

The top of a pie tin is a circle with a diameter of $9.5$ inches. What is the circumference of the top?

A circle has a circumference of $163.28$ inches. Find the diameter.

52 in.

A circle has a circumference of $59.66$ feet. Find the diameter.

A circle has a circumference of $17.27$ meters. Find the diameter.

5.5 m

A circle has a circumference of $80.07$ centimeters. Find the diameter.

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

A circle has a circumference of $150.72$ feet.

24 ft

A circle has a circumference of $251.2$ centimeters.

A circle has a circumference of $40.82$ miles.

6.5 mi

A circle has a circumference of $78.5$ 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.

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.

16 sq. units

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.

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.

30 sq. units

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.

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.

57.5 sq. units

A trapezoid is shown. The bases are labeled 5 and 10, the height is 5.

Two triangles are shown. They appear to be right triangles. The bases are labeled 3, the heights 4, and the longest sides 5.

12 sq. units

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.

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.

67.5 sq. units

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.

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.

89 sq. units

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.

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.

44.81 sq. units

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.

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.

41.12 sq. units

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.

35.13 sq. units

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.

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.

95.625 sq. units

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.

A city park covers one block plus parts of four more blocks, as shown. The block is a square with sides $250$ 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.

187,500 sq. ft

A gift box will be made from a rectangular piece of cardboard measuring $12$ inches by $20$ inches, with squares cut out of the corners of the sides, as shown. The sides of the squares are $3$ 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.

Perry needs to put in a new lawn. His lot is a rectangle with a length of $120$ feet and a width of $100$ feet. The house is rectangular and measures $50$ feet by $40$ feet. His driveway is rectangular and measures $20$ feet by $30$ 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.

9400 sq. ft

Denise is planning to put a deck in her back yard. The deck will be a $20-ft$ by $12-ft$ rectangle with a semicircle of diameter $6$ 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

Area of a Tabletop Yuki bought a drop-leaf kitchen table. The rectangular part of the table is a $1-ft$ by $3-ft$ 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.

ⓐ 6.5325 sq. ft
ⓑ 10.065 sq. ft

Painting Leora wants to paint the nursery in her house. The nursery is an $8-ft$ by $10-ft$ rectangle, and the ceiling is $8$ feet tall. There is a $3-ft$ by $6.5-ft$ door on one wall, a $3-ft$ by $6.5-ft$ closet door on another wall, and one $4-ft$ by $3.5-ft$ 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

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.

Answers will vary.

A circle has a diameter of $14$ feet. Find the area of the circle ⓐ using $3.14$ for $π$ ⓑ using $\frac{22}{7}$ for $π .$ ⓒ 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.

A self-assessment table for students to rate their understanding of using properties of circles and finding areas of irregular figures. Options are 'Confidently', 'With some help', and 'No-I don't get it!' — **Figure 9.32**

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