Solve Geometry Applications: Circles and Irregular Figures

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	\(A=\pi {r}^{2}\) \(A=\pi {(2.5)}^{2}\)
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

1. ⓐ 43.96 in.
2. ⓑ 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

1. ⓐ 53.38 ft
2. ⓑ 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.

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.

Tsopano inu Sikuti pali calculator yomwe imatha kufotokoza izi, koma zigawo zake zimatha kuwerengedwa. Timafuna kuyesera imodzi pansipa, kapena tidzalemba ya ife.

⌨ Chigawo ✎ Kujambula

Mafunso anu a matekinoloje

π Kuchotsa

Kuphunzitsa (40)

Kuyesa zonse pa pepala koyamba. Kusonyeza yankho kuti aone; kutsimikizika amene angatsegulidwe mu wothamanga kwa aliyense gawo.

1. Evaluate \({x}^{2}\) when \(x=5.\)

   Ntchito mwake Chidziwitso

   Kusonyeza yankho

   \(25\)

2. Using \(3.14\) for \(\pi ,\) approximate the (a) circumference and (b) the area of a circle with radius \(8\) inches.

   Ntchito mwake Chidziwitso

   Kusonyeza yankho

   (a) \(50.24\text{in}.\); (b) \(200.96\text{sq}.\text{in}\).

3. Simplify \(\frac{22}{7}{(0.25)}^{2}\) and round to the nearest thousandth.

   Ntchito mwake Chidziwitso

   Kusonyeza yankho

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

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

   Ntchito mwake Chidziwitso

   Kusonyeza yankho

   ⓐ 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	\(A=\pi {r}^{2}\) \(A=\pi {(2.5)}^{2}\)
   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.

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

   Ntchito mwake Chidziwitso

   Kusonyeza yankho

   1. ⓐ 31.4 in.
   2. ⓑ 78.5 sq. in.

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

   Ntchito mwake Chidziwitso

   Kusonyeza yankho

   1. ⓐ 28.26 ft
   2. ⓑ 63.585 sq. ft

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

   Ntchito mwake Chidziwitso

   Kusonyeza yankho

   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.

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

   Ntchito mwake Chidziwitso

   Kusonyeza yankho

   17.27 ft

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

   Ntchito mwake Chidziwitso

   Kusonyeza yankho

   37.68 ft

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

    Ntchito mwake Chidziwitso

    Kusonyeza yankho

    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.

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

    Ntchito mwake Chidziwitso

    Kusonyeza yankho

    30 cm

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

    Ntchito mwake Chidziwitso

    Kusonyeza yankho

    110 ft

13. Find the area of the shaded region.

    Ntchito mwake Chidziwitso

    Kusonyeza yankho

    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.

14. Find the area of each shaded region:

    Ntchito mwake Chidziwitso

    Kusonyeza yankho

    28 sq. units

15. Find the area of each shaded region:

    Ntchito mwake Chidziwitso

    Kusonyeza yankho

    110 sq. units

16. Find the area of the shaded region.

    Ntchito mwake Chidziwitso

    Kusonyeza yankho

    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.

17. Find the area of each shaded region.

    Ntchito mwake Chidziwitso

    Kusonyeza yankho

    36.5 sq. units

18. Find the area of each shaded region.

    Ntchito mwake Chidziwitso

    Kusonyeza yankho

    70 sq. units

19. 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.

    Ntchito mwake Chidziwitso

    Kusonyeza yankho

    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.

20. Find the area:

    Ntchito mwake Chidziwitso

    Kusonyeza yankho

    103.2 sq. units

21. Find the area:

    Ntchito mwake Chidziwitso

    Kusonyeza yankho

    38.24 sq. units

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

    Ntchito mwake Chidziwitso

    Kusonyeza yankho

    1. ⓐ 43.96 in.
    2. ⓑ 153.86 sq. in.

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

    Ntchito mwake Chidziwitso

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

    Ntchito mwake Chidziwitso

    Kusonyeza yankho

    1. ⓐ 53.38 ft
    2. ⓑ 226.865 sq. ft

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

    Ntchito mwake Chidziwitso

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

    Ntchito mwake Chidziwitso

    Kusonyeza yankho

    62.8 ft

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

    Ntchito mwake Chidziwitso

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

    Ntchito mwake Chidziwitso

    Kusonyeza yankho

    37.68 in.

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

    Ntchito mwake Chidziwitso

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

    Ntchito mwake Chidziwitso

    Kusonyeza yankho

    6.908 ft

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

    Ntchito mwake Chidziwitso

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

    Ntchito mwake Chidziwitso

    Kusonyeza yankho

    52 in.

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

    Ntchito mwake Chidziwitso

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

    Ntchito mwake Chidziwitso

    Kusonyeza yankho

    5.5 m

35. A circle has a circumference of \(80.07\) centimeters. Find the diameter.

    Ntchito mwake Chidziwitso

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

    Ntchito mwake Chidziwitso

    Kusonyeza yankho

    24 ft

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

    Ntchito mwake Chidziwitso

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

    Ntchito mwake Chidziwitso

    Kusonyeza yankho

    6.5 mi

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

    Ntchito mwake Chidziwitso

40. 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.

    Ntchito mwake Chidziwitso

    Kusonyeza yankho

    187,500 sq. ft

Kusunga ntchito yanu yokha

M’malo mwake, maakaunti aulere amawonjezera malemba pazophunzira zonse, mnda wa zomwe mwamaliza, mavuto anu omaliza m’malo limodzi, ndi m’bale amene mungafunse za nkhaniyo. Maphunziro a matekinoloje ndi otsegulira kwa aliyense, olembetsa kapena osalembetsa.

Kulembetsa Kulowa

Zithunzi

Pitani pa dzina lililonse lachifaniziro kuti mudziwe tanthauzo lake, chithunzi chake, ndi zimene limatanthauza.

\pipi
Ratio of a circle's circumference to its diameter, 3.14159…\approxapproximately equal
Equal to the precision shown, not exactly.\thetatheta
The usual name for an angle.^\circdegrees
1/360 of a full turn. 180° = π radians.\sinsine, cosine, tangent
Ratios of sides in a right triangle; coordinates on the unit circle.\angle ABCangle, triangle
The angle at B between BA and BC; the triangle with those vertices.\parallelparallel, perpendicular, congruent, similar
Never meet; meet at 90°; identical shape and size; same shape.

Momwe mungagwiritsire ntchito: Solve Geometry Applications: Circles and Irregular Figures

1. Use the properties of circles
2. Find the area of irregular figures

Mafunso omwe anthu amafunsa

Why does every triangle have angles adding to 180°?

Draw a line through one vertex parallel to the opposite side: the two alternate angles equal the other two angles of the triangle, and the three angles at the vertex lie on a straight line. That is Euclid's proof, and it is why the fact holds only on a flat plane.

When do I use the law of sines versus the law of cosines?

Cosines when you know three sides, or two sides and the angle between them. Sines when you know an angle and the side opposite it, plus one more piece.

What is the difference between area and perimeter?

Perimeter is the length of the boundary (one dimension, measured in metres); area is the amount of surface inside (two dimensions, square metres). Doubling every side doubles the perimeter but quadruples the area.

Zigawo za m'nkhaniyi ndi zochokera OpenStax Prealgebra 2e (CC BY-NC-SA 4.0). Kuphatikizapo ndi kufotokozanso pano; zolakwika ndi zathu.

Zambiri pa Geometry

TrianglesPythagorean theoremCirclesPolygonsSolidsCoordinate geometryEuclid's axioms and the structure of proofCongruent and similar trianglesCircle theoremsTransformations: translation, rotation, reflection, dilationSolid geometry: prisms, pyramids, spheres