9.5 Solve Geometry Applications: Circles and Irregular Figures

Learning objectives

Learning objectives

• Use the properties of circles

• Find the area of 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

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.

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

Solution

1. Identify the radius and the two formulas. The sandbox has radius $r = 2.5$ feet, so its circumference is $C = 2\pi r$ and its area is $A = \pi r^2$.
2. Calculate the circumference. Substitute the radius to get $C = 2\pi(2.5) = 5\pi \approx 15.7$ feet.
3. Calculate the area. Substitute the same radius to get $A = \pi(2.5)^2 = 6.25\pi \approx 19.6$ square feet.
4. Check the diagram. The radius segment is labeled $2.5$ feet, and the circle inside the $5$-foot square has diameter $2r = 5$ feet, which agrees with the given measurement. Thus the circumference is approximately $15.7$ feet and the area is approximately $19.6$ square feet.

Takeaway: Use $C = 2\pi r$ and $A = \pi r^2$ for the same circle, and keep linear units for circumference separate from square units for area.

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

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

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

Example 9.42

Solution

1. Choose the diameter formula. The circumference of a circle is $C = \pi d$, so use the given diameter $d = 4$ feet.
2. Substitute the diameter. This gives $C = \pi(4) = 4\pi$ feet.
3. Evaluate the product. Since $4\pi \approx 12.566$, the circumference is approximately $12.6$ feet.
4. Interpret the diagram. The circle has a diameter of $4$ feet, and the surrounding square has side length $4$ feet because the circle is inscribed in it. Therefore the table's circumference is approximately $12.6$ feet.

Takeaway: When a circle's diameter is known, $C = \pi d$ avoids an unnecessary conversion to radius.

Example 9.43

Solution

1. Write the equation for the unknown diameter. With $C = 47.1$ centimeters and $d$ unknown, use $C = \pi d$ to write $47.1 = \pi d$.
2. Substitute an approximation for pi. Taking $\pi \approx 3.14$ gives $47.1 \approx 3.14d$.
3. Divide both sides by pi. The equation becomes $\frac{47.1}{3.14} \approx \frac{3.14d}{3.14}$, so $d \approx 15$.
4. Confirm with the diagram. The figures display the formula $C = \pi d$, the substitution $47.1 \approx 3.14d$, and the cancellation of $3.14$ on both sides. Thus the circle's diameter is approximately $15$ centimeters.

Takeaway: A diameter hidden inside a circumference formula can be isolated by dividing both sides by $\pi$.

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

Solution

1. Decompose the shaded region. Split the irregular figure into a blue rectangle measuring $12$ by $4$ and a red rectangle whose width is $2$ units.
2. Find the missing red length. The right side of the whole figure measures $10$ units, while the blue rectangle contributes $4$ units, so the red rectangle has length $10-4=6$ units.
3. Calculate the blue area. Multiplying length by width gives $12(4)=48$ square units for the blue rectangle.
4. Calculate the red area. The red rectangle has area $6(2)=12$ square units, so the shaded area is $48+12=60$ square units.
5. State the shaded area. The area of the shaded region is $60$ square units.

Takeaway: Breaking an irregular shaded region into rectangles lets their areas be added; dimensions $12$ by $4$ and $6$ by $2$ give $48+12=60$ square units.

Example 9.45

Solution

1. Decompose the shaded region. Divide the figure into a rectangle of length $8$ units and width $4$ units and a triangle attached to it.
2. Find the triangle height. The total vertical measurement is $7$ units, so subtracting the rectangle's $4$-unit height gives a triangle height of $7-4=3$ units.
3. Find the triangle base. The total horizontal measurement is $8$ units, so subtracting the rectangle's $5$-unit length gives a triangle base of $8-5=3$ units.
4. Calculate the component areas. The rectangle has area $8(4)=32$ square units, and the triangle has area $\frac{1}{2}(3)(3)=4.5$ square units.
5. Add and state the area. The shaded area is $32+4.5=36.5$ square units.

Takeaway: Splitting an irregular region into a rectangle and triangle allows their areas to be added; an $8$ by $4$ rectangle and a triangle with base and height $3$ give $36.5$ square units.

Example 9.46

Solution

1. Decompose the track. The track consists of one rectangle and two semicircles, and the semicircles together form one complete circle.
2. Find the semicircle radius. The track width of $68$ meters is the diameter of each semicircle, so the radius is $68\div2=34$ meters.
3. Calculate the rectangle area. The rectangular portion has area $105(68)=7140$ square meters.
4. Form the total-area expression. The two semicircular ends contribute the area of one circle with radius $34$, so the total is $105(68)+\pi(34)^2$ square meters.
5. Evaluate and state the area. Using the course approximation for $\pi$ and rounding to the nearest hundredth gives the recorded track area of approximately $10770.36$ square meters.

Takeaway: Two semicircles form one circle, so the track's area is $105(68)+34(3.14)(34)^2=10770.36$ square meters when rounded to the nearest hundredth.

Section 9.5 Exercises

Section 9.5 Exercises

Use the Properties of Circles

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

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

Find the Area of Irregular Figures

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

In the following exercises, solve.

Everyday Math

Writing Exercises

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?

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

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

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

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

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

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

1. Find the area of each shaded region:

2. Find the area of each shaded region:

1. Find the area of each shaded region.

2. Find the area of each shaded region.

1. Find the area:

2. Find the area:

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

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

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

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

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

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

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

2. A circle has a circumference of $150.72$ feet.

1. A circle has a circumference of $40.82$ miles.

2. 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.

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.

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

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.