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Solve Geometry Applications: Circles and Irregular Figures

Use the properties of circles

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

Try it.

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

Solution

ⓐ 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	\(C=2\pi r\) \(C=2\pi (2.5)\)
Step 5. Solve the equation.	\(C\approx 2(3.14)(2.5)\) \(C\approx 15\text{ft}\)
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	\(A=\text{\pi }{r}^{2}\) \(A=\text{\pi }{(2.5)}^{2}\)
Step 5. Solve the equation.	\(A\approx (3.14){(2.5)}^{2}\) \(A\approx 19.625\ \text{sq. ft}\)
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:

\[C=2\pi r\]

Condensed — the full section is in OpenStax Prealgebra 2e.

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

Try it.

Find the area of the shaded region.

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.

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.

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.

Example

Try it.

Find the area of the shaded region.

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 \(8-5\).

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

The area of the figure is \(36.5\) square units.

Example

Try it.

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.

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.

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.

Condensed — the full section is in OpenStax Prealgebra 2e.

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
- \(d=2r\)
- Circumference: \(C=2\pi r\) or \(C=\pi d\)
- Area: \(A=\pi {r}^{2}\)

Solve Geometry Applications: Circles and Irregular Figures

Use the Properties of Circles

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

Try it.

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

Solution

ⓐ 43.96 in.
ⓑ 153.86 sq. in.

Try it.

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

Try it.

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

Solution

ⓐ 53.38 ft
ⓑ 226.865 sq. ft

Try it.

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

Try it.

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

Solution

62.8 ft

Try it.

A turntable is a circle with diameter of \(10\) inches. What is the circumference of the turntable?

Try it.

A circular saw has a diameter of \(12\) inches. What is the circumference of the saw?

Solution

37.68 in.

Try it.

A round coin has a diameter of \(3\) centimeters. What is the circumference of the coin?

Try it.

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

Solution

6.908 ft

Try it.

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

Try it.

A circle has a circumference of \(163.28\) inches. Find the diameter.

Solution

52 in.

Try it.

A circle has a circumference of \(59.66\) feet. Find the diameter.

Try it.

A circle has a circumference of \(17.27\) meters. Find the diameter.

Solution

5.5 m

Try it.

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.

Try it.

A circle has a circumference of \(150.72\) feet.

Solution

24 ft

Try it.

A circle has a circumference of \(251.2\) centimeters.

Try it.

A circle has a circumference of \(40.82\) miles.

Solution

6.5 mi

Try it.

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.

Try it.

Solution

16 sq. units

Try it.

Solution

30 sq. units

Try it.

Solution

57.5 sq. units

Try it.

Solution

12 sq. units

Try it.

Solution

67.5 sq. units

Try it.

Solution

89 sq. units

Try it.

Solution

44.81 sq. units

Try it.

Solution

41.12 sq. units

Try it.

Solution

35.13 sq. units

Try it.

Solution

95.625 sq. units

Try it.

In the following exercises, solve.

Try it.

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.

Solution

187,500 sq. ft

Try it.

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.

Try it.

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.

Solution

9400 sq. ft

Try it.

Denise is planning to put a deck in her back yard. The deck will be a \(\text{20-ft}\) by \(\text{12-ft}\) rectangle with a semicircle of diameter \(6\) feet, as shown below. Find the area of the deck.

Condensed — the full section is in OpenStax Prealgebra 2e.

Practice (40)

Try each one on paper first. Reveal the answer to check; verified ones can be opened in the solver for every step.

Evaluate \({x}^{2}\) when \(x=5.\)
If you missed this problem, review .

Otkrij odgovor

\(25\)
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 .

Otkrij odgovor

(a) \(50.24\text{in}.\); (b) \(200.96\text{sq}.\text{in}\).
Simplify \(\frac{22}{7}{(0.25)}^{2}\) and round to the nearest thousandth.
If you missed this problem, review .

Otkrij odgovor

\(\text{0}\text{.196}\)

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

Otkrij odgovor

ⓐ 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	\(C=2\pi r\) \(C=2\pi (2.5)\)
Step 5. Solve the equation.	\(C\approx 2(3.14)(2.5)\) \(C\approx 15\text{ft}\)
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	\(A=\text{\pi }{r}^{2}\) \(A=\text{\pi }{(2.5)}^{2}\)
Step 5. Solve the equation.	\(A\approx (3.14){(2.5)}^{2}\) \(A\approx 19.625\ \text{sq. ft}\)
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.

A circular mirror has radius of \(5\) inches. Find the ⓐ circumference and ⓑ area of the mirror.
Otkrij odgovor
1. ⓐ 31.4 in.
2. ⓑ 78.5 sq. in.
A circular spa has radius of \(4.5\) feet. Find the ⓐ circumference and ⓑ area of the spa.
Otkrij odgovor
1. ⓐ 28.26 ft
2. ⓑ 63.585 sq. ft

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

Otkrij odgovor

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.	\(C=\pi d\) \(C=\pi (4)\)
Step 5. Solve the equation, using 3.14 for \(\pi .\)	\(C\approx (3.14)(4)\) \(C\approx 12.56\ \text{feet}\)
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.

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

Otkrij odgovor

17.27 ft
If the diameter of a circular trampoline is \(12\) feet, what is its circumference?

Otkrij odgovor

37.68 ft

Find the diameter of a circle with a circumference of \(47.1\) centimeters.

Otkrij odgovor

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 \(\pi\).
Step 5. Solve.
Step 6. Check: \(47.1\overset{?}{=}(3.14)(15)\) \(47.1=47.1✓\)
Step 7. Answer the question.	The diameter of the circle is approximately 15 centimeters.

Find the diameter of a circle with circumference of \(94.2\) centimeters.

Otkrij odgovor

30 cm
Find the diameter of a circle with circumference of \(345.4\) feet.

Otkrij odgovor

110 ft
Find the area of the shaded region.

Otkrij odgovor

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.
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.
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.
Find the area of each shaded region:

Otkrij odgovor

28 sq. units
Find the area of each shaded region:

Otkrij odgovor

110 sq. units
Find the area of the shaded region.

Otkrij odgovor

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 \(8-5\).
Now we can add the areas to find the area of the irregular figure.

The area of the figure is \(36.5\) square units.
Find the area of each shaded region.

Otkrij odgovor

36.5 sq. units
Find the area of each shaded region.

Otkrij odgovor

70 sq. units
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.

Otkrij odgovor

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.
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.
Find the area:

Otkrij odgovor

103.2 sq. units
Find the area:

Otkrij odgovor

38.24 sq. units
The lid of a paint bucket is a circle with radius \(7\) inches. Find the ⓐ circumference and ⓑ area of the lid.
Otkrij odgovor
1. ⓐ 43.96 in.
2. ⓑ 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.
Otkrij odgovor
1. ⓐ 53.38 ft
2. ⓑ 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?

Otkrij odgovor

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?

Otkrij odgovor

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?

Otkrij odgovor

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.

Otkrij odgovor

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.

Otkrij odgovor

5.5 m
A circle has a circumference of \(80.07\) centimeters. Find the diameter.
A circle has a circumference of \(150.72\) feet.

Otkrij odgovor

24 ft
A circle has a circumference of \(251.2\) centimeters.
A circle has a circumference of \(40.82\) miles.

Otkrij odgovor

6.5 mi
A circle has a circumference of \(78.5\) inches.
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.

Otkrij odgovor

187,500 sq. ft

Symbols used here

\pi

pi
Ratio of a circle's circumference to its diameter, 3.14159…

\approx

approximately equal
Equal to the precision shown, not exactly.

\pm

plus or minus
Both signs at once: x = 3 ± 2 means 5 and 1.

\neq

not equal
The two sides are different.

\leq,\ \geq

less/greater than or equal
Inequalities that allow equality; < and > exclude it.

\sqrt{x},\ \sqrt[n]{x}

square root, n-th root
The non-negative number whose square (n-th power) is x.

a \bmod n

remainder
What is left after dividing a by n.

per cent
Per hundred: 15% = 15/100.

a : b,\ \frac{a}{b}

ratio, fraction
a for every b; a divided by b.

How to: Solve Geometry Applications: Circles and Irregular Figures

Use the properties of circles
Find the area of irregular figures

Questions people ask

Why does the order of operations matter?

Because 2 + 3 × 4 would otherwise be two different numbers. The convention (brackets, exponents, multiplication and division, addition and subtraction) exists so every reader gets the same value from the same expression.

How do I check an arithmetic answer?

Estimate first (round every number and compute roughly), then compare. If the estimate and the exact answer disagree by more than a little, one of them is wrong. The solver shows every operation, so you can find which line went astray.

Why are fractions harder than decimals?

They are not harder, they are more exact: 1/3 is a precise number, 0.333 is an approximation. Fractions need a common denominator to add, which is the one extra step people trip on.

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Parts of this page are adapted from OpenStax Prealgebra 2e (CC BY-NC-SA 4.0). Condensed and re-explained here; errors are ours.

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