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1.3 Solve Geometry Applications: Circles and Irregular Figures

Learning Objectives

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

  • Use the properties of circles
  • Find the area of irregular figures

How To

Problem Solving Strategy for Geometry Applications

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. 

Step 2. Identify what you are looking for.

Step 3. Name what you are looking for. Choose a variable to represent that quantity. 

Step 4. Translate into an equation by writing the appropriate formula or model for the situation. Substitute in the given information.

Step 5. Solve the equation using good algebra techniques.

Step 6. Check the answer in the problem and make sure it makes sense.

Step 7. Answer the question with a complete sentence.

Use the Properties of Circles

Properties of Circles

A circle with a line drawn through the widest part through the center labeled d for diameter. Each segment from the center is half of d, or the radius r.
  • r is the length of the radius
  • d is the length of the diameter, [latex]d=2r[/latex]
  • Circumference is the perimeter of a circle.
  • The formula for circumference is [latex]C=2\pi r[/latex]
  • The formula for area of a circle is [latex]A=\pi r^2[/latex]

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

Finding Diameter of a Circle Given Circumference

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

Show Solution
    1. Read: Read the problem carefully.
    2. Identify: We need to find the diameter.
    3. Formula: Recall that Circumference [latex]C=\pi d[/latex], where [latex]d[/latex] is the diameter.
    4. Substitute: Plug in the given circumference: [latex]47.1 = \pi d[/latex]
    5. Solve: Divide both sides by [latex]\pi[/latex]: [latex]d = 47.1 / \pi[/latex], with [latex](\pi \approx 3.14)[/latex]
    6. Calculate: [latex]d\approx 47.1/3.14\approx 15[/latex] cm
    7. Answer: The diameter of the circle is approximately 15 centimeters.

Try It

A circular mirror has radius of 5 inches. Find the (a) circumference and (b) area of the mirror.

Show Solution

(a) Circumference = 31.4 in.
(b) Area = 78.5 square in.

Try It

A circular spa has radius of 4.5 feet. Find the (a) circumference and (b) area of the spa.

Show Solution

(a) Circumference = 28.27 ft.
(b) Area = 63.62 square ft.

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

[latex]C = 2\pi r[/latex]

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

Explanation Steps
Formula [latex]C = 2\pi r[/latex]
Using the commutative property, we get [latex]C = \pi \cdot 2r[/latex]
Then substituting d = 2r [latex]C = \pi \cdot d   [/latex]
So [latex]C = \pi d   [/latex]

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

Finding Circumference

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

Show Solution
  1. Explanation Steps
    Step 1. Read the problem. Draw the figure and label it with the given information. An overhead view of a circular table has a diameter of four feet
    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=πdC=πd C=π(4)C=π(4)
     

    Step 5. Solve the equation, using 3.14 for π.

    		  

    C≈(3.14)(4)C≈(3.14)(4) C≈12.56feetC≈12.56feet
     

    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.

    		 A circle with diameter of 4 ft inscribed in a square with side 4 ft.
    Step 7. Answer the question. The circumference of the table is 12.56 feet.

Try It

Find the circumference of a circular fire pit whose diameter is 5.5 feet.

Show Solution

Circumference = 17.27 feet

Try It

If the diameter of a circular trampoline is 12 feet, what is its circumference?

Show Solution

Circumference = 37.70 feet

Finding Diameter of a Circle Given Circumference

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

Show Solution
Explanation Steps
Step 1. Read the problem. Draw the figure and label it with the given information. Circle with diameter d and C=47.1 cm.
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.π		 [latex]C = \pi d[/latex]

[latex]47.1 \approx 3.14 d[/latex]
Step 5. Solve. [latex]\frac {47.1}{3.14} \approx \frac{3.14 d}{3.14}[/latex]

[latex]15 \approx d[/latex]
Step 6. Check:
[latex]C= \pi d[/latex][latex]47.1 \stackrel{?}{=} (3.14)(15)[/latex][latex]47.1 = 47.1[/latex] ✓
Step 7. Answer the question. The diameter of the circle is approximately 15 centimeters.

Try It

Find the diameter of a circle with circumference of 94.2 centimeters.

Show Solution

Diameter = 30 centimeters

Try It

Find the diameter of a circle with circumference of 345.4 feet.

Show Solution

Diameter = 110 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.

 

Finding Area of Piecewise Rectangular Region

Find the area of the shaded region.


A horizontal rectangle (12×4) attached to a vertical rectangle (10×2) forming an L‑shaped figure.
Show Solution

The given figure is irregular, but we can break it into two rectangles. The area of the shaded region will be the sum of the areas of both rectangles.


Attached horizontal and vertical rectangles; top labeled 12, horizontal side 4, right side 10, and vertical width 2.

The blue rectangle has a width of 12 and a length of 4. The red rectangle has a width of 2, but its length is not labeled. The right side of the figure is the length of the red rectangle plus the length of the blue rectangle. Since the right side of the blue rectangle is 4 units long, the length of the red rectangle must be 6 units.

Blue horizontal rectangle attached to red vertical one; top 12, blue side 4, full side 10 (4 blue, 6 red), red width 2.

Text shows area steps: A(figure) = A(blue) + A(red) = blue bh + red bh = blue 12×4 + red 2×6 = blue 48 + red 12 = 60.

The area of the figure is 60 square units.

Is there another way to split this figure into two rectangles? Try it, and make sure you get the same area.

Try It

Find the area of each shaded region:


Horizontal rectangle (8x2) attached to vertical rectangle (6x3).

Show Solution

Area = 28 square units

Try It 

Find the area of each shaded region:


A horizontal rectangle attached to a vertical one; top 14, horizontal width 5, side 10, missing width 6.

Show Solution

Area = 110 square units

Finding Area of Irregular Region

Find the area of the shaded region.


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.

Show Solution

We can break this irregular figure into a triangle and rectangle. The area of the figure will be the sum of the areas of triangle and rectangle. The rectangle has a length of 8 units and a width of 4 units.

We need to find the base and height of the triangle.

Since both sides of the rectangle are 4, the vertical side of the triangle is 3, which is 7 − 4.

The length of the rectangle is 8, so the base of the triangle will be 3, which is 8 − 4.


The same shape from above, showing the 2 different shapes: rectangle is 4x8 and the triangle has base 3 and height 3.

Now we can add the areas to find the area of the irregular figure.


Text shows area steps: A(figure) = A(rec) + A(Δ) = lw + ½ bh = 8×4 + ½ x3×3 = 32 +  4.5 = 36.5 sq units.

The area of the figure is 36.5 square units.

Try It

Find the area of each shaded region.


Rectangle with triangle on lower right; rectangle base 8, height 4; triangle starts 1 from top and has top length 3.

Show Solution

Area = 36.5 square units

Try It

Find the area of each shaded region.


A 5x12 rectangle connected to an isosceles triangle with height 4 and base split into two 2.5‑unit segments.

Show Solution

Area = 70 square units

Finding Area of Irregular Region

A high school track is shaped like a rectangle with a semi-circle (half a circle) on each end. The rectangle has length 105 meters and width 68 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..

Show Solution

We will break the figure into a rectangle and two semi-circles. The area of the figure will be the sum of the areas of the rectangle and the semicircles.

A 68x105 rectangle with semicircles on each side. The diameter of the semicircles are adjacent to the 68 m side of the rectangle.

The rectangle has a length of 105 m and a width of 68 m. The semi-circles have a diameter of 68 m, so each has a radius of 34 m.

Here's how to solve this problem:

  1. Area of the rectangle: Length × Width = 105 m × 68 m = 7140 m²
  2. Area of the two semi-circles: Two semi-circles make a full circle. The diameter of the circle is equal to the width of the rectangle (68 m). So, the radius is 68 m / 2 = 34 m. Area of a circle = πr² = π × (34 m)² ≈ 3631.68 m²
  3. Total Area: Area of rectangle + Area of circle ≈ 7140 m² + 3631.68 m² ≈ 10771.68 m²

Therefore, the area enclosed by the track is approximately 10771.68 square meters.

Area steps: A(figure) = A(rect) + A( 2 semi cirlces) = bh + 2(½πr²) ≈ 105×68 +  2(½·3.14·34²) = 7140 + 3629.84 ≈ 10,769.84 m².

Try It

Find the area:


A 9x15 rectangle with a missing section; a red circle of equal height (9) attaches to the missing semi-circle.

Show Solution

Area = 103.2 square units

Try It

Find the area:


Two trapezoids with a semicircle at the top. The base of the semicircle is 5.2. The height of the trapezoids is 6.5. The combined base of the trapezoids is 3.3.

Show Solution

Area = 38.2 square units

Key Concepts

Problem Solving Strategy for Geometry Applications

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. 

Step 2. Identify what you are looking for.

Step 3. Name what you are looking for. Choose a variable to represent that quantity. 

Step 4. Translate into an equation by writing the appropriate formula or model for the situation. Substitute in the given information.

Step 5. Solve the equation using good algebra techniques.

Step 6. Check the answer in the problem and make sure it makes sense.

Step 7. Answer the question with a complete sentence.

Properties of Circles

A circle with a line drawn through the widest part through the center labeled d for diameter. Each segment from the center is half of d, or the radius r.
  • r is the length of the radius, d is the length of the diameter, [latex]d = 2r[/latex]
  • Circumference is the perimeter of a circle. The formula for circumference is [latex]C = 2\pi r[/latex]
  • The formula for area of a circle is [latex]A = \pi r^2[/latex]

 

Section Exercises

Use the Properties of Circles

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

1. The lid of a paint bucket is a circle with radius 7.

Find the (a) circumference and (b) area of the lid.

Show Solution to (a)
43.96 in.  
Show Solution to (b)
153.86 sq. in. 

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

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

Show Solution to (a)
53.38 ft
Show Solution to (b)
226.865 sq. ft  

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

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

Show Solution
62.8 ft

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

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

Show Solution
37.68 in. 

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

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

Show Solution
6.908 ft

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

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

Show Solution
52 in.

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

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

Show Solution
5.5 m

14. 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.

15. A circle has a circumference of 150.72 feet. 

Show Solution
24 ft

16. A circle has a circumference of 251.2 centimeters. 

17. A circle has a circumference of 40.82 miles.

Show Solution
6.5 mi  

18. 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.

19.


A horizontal rectangle (2x6) attached to a vertical rectangle (4x2).
Show Solution
16 sq. units

 

20.


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.

 

21.


Sideways U‑shaped figure; top 6, left side 6, inner horizontal 3, and two right vertical segments each labeled 2.
Show Solution
30 sq. units 

22.


U‑shaped figure with base 7, right side 5, top horizontals 3 each, and inner vertical segment also labeled 3.

23.


A 4x10 rectangle with triangle on bottom left. The triangle begins 3 from the top and the combined base of both shapes is 9.
Show Solution
27.5 sq. units 

24.


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

25.


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

26.


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.

27.


2 trapezoids. The base is 10. The height of one trapezoid is 2. The horizontal and vertical sides are all 5.
Show Solution
67.5 sq. units 

28.


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.

29.


Shape containing a 8x8 rectangle, another 2x13 rectangle, and a triangle with height (13-8) and base (8-2).
Show Solution
89 sq. units 

30.


Shape containing a 7x6 rectangle, another 1x12 rectangle, and a triangle with height (12-7) and base (6-1).

31.


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.
Show Solution
44.81 sq. units 

32.


A 6x10 rectangle with attached semicircle; the semicircle diameter is (10-5-2).

33


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.
Show Solution
41.12 sq. units 

34.


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.


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.
Show Solution
35.13 sq. units 

36.


A trapezoid with a semi-circle attached to the top. The diameter of the circle, & top of the trapezoid is 8. The trapezoid has h=6 and B=13.

37.


A 6×11 rectangle with a triangle of height 5 and base 4 on the top left, and a circle of diameter 5 on the top right.
Show Solution
95.625 sq. units 

38.


Trapezoid (top 6, bottom 9, height 7) with a triangle on top (height 5, base 6) and a circle of diameter 4 attached above.

 

In the following exercises, solve.

39. 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.

Show Solution
187, 500 sq. ft.

40. 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.

 

41. 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.

Show Solution
9400 sq. ft 

42. 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.

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

43. 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. (a) Find the area of the table with one leaf up. (b) Find the area of the table with both leaves up.

The table with a rectangular portion attached to a semi-circular portion. There is another semi-circular leaf folded down on the other side of the rectangle.

Show Solution to (a)
6.5325 sq. ft 
Show Solution to (b)
10.065 sq. ft 

44. 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

45. 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.


Vertical rectangle (width 3, left side 6) attached to horizontal one (bottom 9, width 3, top 6); gap above it labeled 3.

46. A circle has a diameter of 14 feet. Find the area of the circle (a) using 3.14 for π,  and (b) using [latex]\frac{22}{7}[/latex] or [latex]\pi[/latex]. (c) Which calculation do you prefer? Why?

Glossary

irregular figure

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.

 

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1.3 Solve Geometry Applications: Circles and Irregular Figures Copyright © by LOUIS: The Louisiana Library Network is licensed under a Creative Commons Attribution 4.0 International License, except where otherwise noted.

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