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Mathematics LibreTexts

9.5: Solve Geometry Applications: Circles and Irregular Figures

  • Page ID
    5009
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    Skills to Develop

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

    be prepared!

    Before you get started, take this readiness quiz.

    1. Evaluate x2 when x = 5. If you missed this problem, review Example 2.15.
    2. Using 3.14 for \(\pi\), approximate the (a) circumference and (b) the area of a circle with radius 8 inches. If you missed this problem, review Example 5.39.
    3. Simplify \(\frac{22}{7}\)(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

    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

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

    Definition: 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.

    • r is the length of the radius
    • d is the length of the diameter
    • d = 2r
    • Circumference is the perimeter of a circle. The formula for circumference is C = 2\(\pi\)r
    • The formula for area of a circle is A = \(\pi\)r2

    Remember, that we approximate \(\pi\) 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 \(\pi\) 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 \(\pi\) key uses more than two decimal places.

    Example 9.41:

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

    Solution

    (a)

    Step 1. Read the problem. Draw the figure and label it with the given information. CNX_BMath_Figure_09_05_029_img-01.png
    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. $$\begin{split} C &= 2 \pi r \\ C &= 2 \pi (2.5) \end{split}$$
    Step 5. Solve the equation. $$\begin{split} C &\approx 2(3.14)(2.5) \\ C &\approx 15\; ft \end{split}$$
    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.

    CNX_BMath_Figure_09_05_029_img-02.png

    Step 7. Answer the question. The circumference of the sandbox is 15.7 feet.

    (b)

    Step 1. Read the problem. Draw the figure and label it with the given information. CNX_BMath_Figure_09_05_029_img-01.png
    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. $$\begin{split} A &= \pi r^{2} \\ A &= \pi (2.5)^{2} \end{split}$$
    Step 5. Solve the equation. $$\begin{split} A &\approx (3.14)(2.5)^{2} \\ A &\approx 19.625\; sq.\; ft \end{split}$$
    Step 6. Check. Does this answer make sense? Yes. If we draw a square around the circle, its sides would be 5 ft, as shown in part (a). 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.

    Exercise 9.81:

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

    Exercise 9.82:

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

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

    $$C = 2 \pi r$$

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

    $$\begin{split} C &= 2 \pi r \\ Using\; the\; commutative\; property,\; we\; get \qquad C &= \pi \cdot 2r \\ Then\; substituting\; d = 2r \qquad C &= \pi \cdot d \\ So \qquad C &= \pi d \end{split}$$

    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?

    Solution

    Step 1. Read the problem. Draw the figure and label it with the given information. CNX_BMath_Figure_09_05_032_img-01.png
    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. $$\begin{split} C &= \pi d \\ C &= \pi (4) \end{split}$$
    Step 5. Solve the equation, using 3.14 for \(\pi\). $$\begin{split} C &\approx (3.14)(4) \\ C &\approx 12.56\; ft \end{split}$$
    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.

    CNX_BMath_Figure_09_05_032_img-02.png

    Step 7. Answer the question. The diameter of the table is 12.56 square feet.

    Exercise 9.83:

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

    Exercise 9.84:

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

    Example 9.43:

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

    Solution

    Step 1. Read the problem. Draw the figure and label it with the given information. CNX_BMath_Figure_09_05_033_img-01.png
    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 \(\pi\). $$\begin{split} C &= \pi d \\ 47.1 &\approx 3.14d \end{split}$$
    Step 5. Solve. $$\begin{split} \frac{47.11}{3.14} &\approx \frac{3.14d}{3.14} \\ 15 &\approx d \end{split}$$
    Step 6. Check. $$\begin{split} 47.1 &\stackrel{?}{=} (3.14)(15) \\ 47.1 &= 47.1\; \checkmark \end{split}$$
    Step 7. Answer the question. The diameter of the circle is approximately 15 centimeters.

    Exercise 9.85:

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

    Exercise 9.86:

    Find the diameter of a circle with circumference of 345.4 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.

    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.

    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.

    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.

    An image of a blue horizontal rectangle attached to a red vertical rectangle is shown. The top is labeled 12, the side of the blue rectangle is labeled 4. The whole side is labeled 10, the blue portion is labeled 4 and the red portion is labeled 6. The width of the red rectangle is labeled 2.

    $$\begin{split} A_{figure} &= A_{\textcolor{blue}{rectangle}} + A_{\textcolor{red}{rectangle}} \\ A_{figure} &= \textcolor{blue}{bh} + \textcolor{red}{bh} \\ A_{figure} &= \textcolor{blue}{12 \cdot 4} + \textcolor{red}{2 \cdot 6} \\ A_{figure} &= \textcolor{blue}{48} + \textcolor{red}{12} \\ A_{figure} &= 60 \end{split}$$

    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.

    Exercise 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.

    Exercise 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.

    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.

    A geometric shape is shown. It is a blue rectangle with a red 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. The right side of the rectangle is labeled 4. The right side and bottom of the triangle are labeled 3.

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

    $$\begin{split} A_{figure} &= A_{\textcolor{blue}{rectangle}} + A_{\textcolor{red}{triangle}} \\ A_{figure} &= \textcolor{blue}{lw} + \frac{1}{2} \textcolor{red}{bh} \\ A_{figure} &= \textcolor{blue}{8 \cdot 4} + \frac{1}{2} \cdot \textcolor{red}{3 \cdot 3} \\ A_{figure} &= \textcolor{blue}{32} + \textcolor{red}{4.5} \\ A_{figure} &= 3.65\; sq.\; units \end{split}$$

    The area of the figure is 36.5 square units.

    Exercise 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.

    Exercise 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 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.

    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 blue geometric shape is shown. It looks like a rectangle with a semi-circle attached to each side. The base of the rectangle is labeled 105 m. The height of the rectangle and diameter of the circle on the left is labeled 68 m.

    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.

    $$\begin{split} A_{figure} &= A_{\textcolor{blue}{rectangle}} + A_{\textcolor{red}{semicircles}} \\ A_{figure} &= \textcolor{blue}{bh} + \textcolor{red}{2\left(\dfrac{1}{2} \pi \cdot r^{2} \right)} \\ A_{figure} &\approx \textcolor{blue}{105 \cdot 68} + \textcolor{red}{2 \left(\dfrac{1}{2} \cdot 3.14 \cdot 34^{2}\right)} \\ A_{figure} &\approx \textcolor{blue}{7140} + \textcolor{red}{3629.84} \\ A_{figure} &\approx 10,769.84\; square\; meters \end{split}$$

    Exercise 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.

    Exercise 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.

    Practice Makes Perfect

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

    1. A circle has a circumference of 150.72 feet.
    2. A circle has a circumference of 251.2 centimeters.
    3. A circle has a circumference of 40.82 miles.
    4. 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.

    1. 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.
    2. 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.
    3. 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.
    4. 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.
    5. 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.
    6. A trapezoid is shown. The bases are labeled 5 and 10, the height is 5.
    7. Two triangles are shown. They appear to be right triangles. The bases are labeled 3, the heights 4, and the longest sides 5.
    8. 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.
    9. 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.
    10. 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.
    11. 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.
    12. 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.
    13. 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.
    14. 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.
    15. 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.
    16. 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.
    17. 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.
    18. 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.
    19. 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.
    20. 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.

    1. 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.

    1. 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.

    1. 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.

    1. 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

    1. 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.

    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.

    1. 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

    1. 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.

    1. A circle has a diameter of 14 feet. Find the area of the circle (a) using 3.14 for \(\pi\) (b) using \(\frac{22}{7}\) for \(\pi\). (c) Which calculation to do prefer? Why?

    Self Check

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

    CNX_BMath_Figure_AppB_055.jpg

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

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