Circle Geometry: Arc Length and Sector Area Practice Questions — ACT — Page 2

Question 21Question →

A decorative circular stained-glass window has a radius of $10$ inches. A specific colored section of the window forms a sector with a central angle measuring $108^\circ$ . What is the total perimeter, in inches, of this stained-glass sector?

6\pi

3\pi + 20

6\pi + 10

6\pi + 20

30\pi + 20

Show answer & explanation

Answer: $6\pi + 20$

Answer

The total perimeter of the sector is

6\pi + 20

inches.

The arc length for a central angle of

108^\circ

in a circle of radius

10

is

\frac{108}{360} \times 2\pi(10) = 6\pi

inches. Adding the two bounding radii (

10 + 10 = 20

) yields the full sector perimeter of

6\pi + 20

inches.

Step-by-Step Solution

1

Calculate the arc length of the sector

Arc length =

\frac{108^\circ}{360^\circ} \cdot 2\pi(10) = \frac{3}{10} \cdot 20\pi = 6\pi

inches.

The curved boundary of a sector is a fraction of the circle's full circumference defined by the central angle ratio.

2

Calculate the total perimeter of the sector

Perimeter = Arc length +

2 \cdot \text{radius} = 6\pi + 2(10) = 6\pi + 20

inches.

A sector's perimeter consists of its outer curved arc plus its two straight radial edges.

Key Concept

Perimeter of a Sector

Estimated Time:1m 0s

Question 22Question →

A robotic arm mounted on a flat work surface pivots through a central angle of $\theta$ degrees, sweeping out a sector of a circle with a radius of $15$ centimeters. If the total perimeter of the swept sector is $(30 + 5\pi)$ centimeters, what is the value of $\theta$ ?

8^\circ

30^\circ

45^\circ

60^\circ

120^\circ

Show answer & explanation

Answer: $60^\circ$

Answer

The measure of the central angle

\theta

is

60^\circ

.

The perimeter of a sector consists of two straight radii of length

r

plus the arc length

s

, so

\text{Perimeter} = 2r + s

. Given

r = 15

cm and a perimeter of

(30 + 5\pi)

cm, we find

2(15) + s = 30 + 5\pi

, which means

s = 5\pi

cm. Using the arc length formula

s = \frac{\theta}{360^\circ} \times 2\pi r

, we set

5\pi = \frac{\theta}{360^\circ} \times 30\pi

. Simplifying yields

\frac{\theta}{12^\circ} = 5

, giving

\theta = 60^\circ

.

Step-by-Step Solution

1

Express the perimeter of a sector in terms of radius

r

and arc length

s

.

\text{Perimeter} = 2r + s

A sector's boundary consists of two straight radii and one curved arc.

2

Substitute

r = 15

cm and the given perimeter

(30 + 5\pi)

cm to solve for the arc length

s

.

30 + s = 30 + 5\pi \implies s = 5\pi \text{ cm}

Subtracting the combined length of the two straight edges (

2 \times 15 = 30

) isolates the arc length.

3

Set up the arc length formula using degrees:

s = \frac{\theta}{360^\circ} \times 2\pi r

.

5\pi = \frac{\theta}{360^\circ} \times 2\pi(15) = \frac{30\pi \theta}{360^\circ} = \frac{\pi \theta}{12^\circ}

The arc length represents the fraction

\frac{\theta}{360^\circ}

of the circle's total circumference

2\pi r

.

4

Solve the equation for

\theta

.

5\pi = \frac{\pi \theta}{12^\circ} \implies 5 = \frac{\theta}{12^\circ} \implies \theta = 60^\circ

Dividing both sides by

\pi

and multiplying by

12^\circ

isolates

\theta

.

Key Concept

Perimeter of a Sector and Arc Length

Estimated Time:1m 15s

Question 23Question →

A garden bed is designed in the shape of a sector of a circle with a central angle measuring $120^\circ$ . If the total perimeter of the sector-shaped garden bed is $24 + 8\pi$ feet, what is the area of the garden bed, in square feet?

8\pi

24\pi

48\pi

96\pi

144\pi

Show answer & explanation

Answer: $48\pi$

Answer

The area of the garden bed is

48\pi

square feet.

The perimeter of a sector is given by

2r + \frac{\theta}{360^\circ}(2\pi r)

. Substituting

\theta = 120^\circ

gives

2r + \frac{2\pi r}{3} = 24 + 8\pi

, which simplifies to

r = 12

feet. Using the sector area formula

\frac{\theta}{360^\circ}\pi r^2

with

r = 12

yields

\frac{1}{3}\pi(144) = 48\pi

square feet.

Step-by-Step Solution

1

Set up the formula for the perimeter of a sector.

Perimeter

= 2r + \text{Arc Length} = 2r + \left(\frac{\theta}{360^\circ}\right)(2\pi r)

A sector's perimeter consists of two straight radii and the curved arc boundary.

2

Substitute

\theta = 120^\circ

into the perimeter expression and solve for the radius

r

.

2r + \left(\frac{120^\circ}{360^\circ}\right)(2\pi r) = 24 + 8\pi \implies 2r + \frac{2\pi r}{3} = 24 + 8\pi \implies r\left(2 + \frac{2\pi}{3}\right) = 12\left(2 + \frac{2\pi}{3}\right) \implies r = 12

feet

Equating the algebraic expression to the given perimeter allows finding the circle radius.

3

Calculate the sector area using

r = 12

feet and

\theta = 120^\circ

.

Area

= \left(\frac{120^\circ}{360^\circ}\right) \pi r^2 = \frac{1}{3} \pi (12)^2 = \frac{144\pi}{3} = 48\pi

square feet

The area of a sector is the fraction of the total circle's area defined by the ratio of central angle to

360^\circ

.

Key Concept

Perimeter and Area of a Circle Sector

Estimated Time:1m 0s

Question 24Question →

A circular stage turntable has a radius of $9$ meters. A wedge-shaped performance zone on the turntable is formed by a sector with a central angle measuring $100^\circ$ . A safety border is installed along the entire boundary of this performance zone, which consists of its curved arc and both straight radius edges. What is the total length, in meters, of the safety border?

5\pi

5\pi + 9

5\pi + 18

2.5\pi + 18

22.5\pi

Show answer & explanation

Answer: $5\pi + 18$

Answer

The total length of the safety border is

5\pi + 18

meters.

To find the total length of the border surrounding the sector, calculate the arc length of the sector using

s = \frac{\theta}{360^\circ} \times 2\pi r = \frac{100}{360} \times 18\pi = 5\pi

meters. Then, add the two straight radial sides (

2r = 2 \times 9 = 18

meters) to get the complete perimeter of

5\pi + 18

meters.

Step-by-Step Solution

1

Calculate the arc length (

s

) of the circular sector.

s = \frac{100^\circ}{360^\circ} \times 2\pi(9) = \frac{5}{18} \times 18\pi = 5\pi

meters.

The arc length formula for a sector measured in degrees is

s = \frac{\theta}{360^\circ} \times 2\pi r

.

2

Calculate the total perimeter (

P

) of the sector.

P = s + 2r = 5\pi + 2(9) = 5\pi + 18

meters.

The boundary of a sector includes the curved arc length plus two straight radial sides.

Key Concept

Perimeter of a Circle Sector

Estimated Time:1m 0s

Question 25Question →

The tip of a pendulum swings through a central angle measuring $\frac{\pi}{3}$ radians, tracing an arc length of $4\pi$ centimeters. What is the area, in square centimeters, of the sector swept out by the pendulum?

12\pi

24\pi

36\pi

48\pi

144\pi

Show answer & explanation

Answer: $24\pi$

Answer

24\pi

square centimeters

To find the area of the sector, first determine the radius of the pendulum's circular path using the arc length formula

s = r\theta

. Substituting

s = 4\pi

and

\theta = \frac{\pi}{3}

gives

r = 12

cm. Then, substitute the radius and central angle into the sector area formula

A = \frac{1}{2}r^2\theta = \frac{1}{2}(12)^2\left(\frac{\pi}{3}\right) = 24\pi

square centimeters.

Step-by-Step Solution

1

Calculate the radius of the circle using the arc length formula in radians.

Using

s = r\theta

with

s = 4\pi

and

\theta = \frac{\pi}{3}

gives

4\pi = r\left(\frac{\pi}{3}\right) \implies r = 12

cm.

The radius is required to calculate the sector area.

2

Calculate the area of the sector using the radian sector area formula.

A = \frac{1}{2}r^2\theta = \frac{1}{2}(12)^2\left(\frac{\pi}{3}\right) = \frac{1}{2}(144)\left(\frac{\pi}{3}\right) = 24\pi

square centimeters.

Multiplying half the squared radius by the radian measure of the central angle yields the sector area.

Key Concept

Arc Length and Sector Area in Radians

Estimated Time:1m 0s

Question 26Question →

A circular section of a community park is bounded by two straight footpaths meeting at the park's center at a $75^\circ$ angle, and an outer curved walking trail. If the length of the outer curved walking trail is $5\pi$ meters, what is the area, in square meters, of this section of the park?

25\pi

30\pi

60\pi

120\pi

144\pi

Show answer & explanation

Answer: $30\pi$

Answer

The area of this section of the park is

30\pi

square meters.

The central angle of

75^\circ

represents

\frac{75}{360} = \frac{5}{24}

of the circle. Setting the arc length

5\pi = \frac{5}{24} (2\pi r)

gives

\frac{5\pi r}{12} = 5\pi

, which simplifies to

r = 12

meters. The sector area is

\frac{5}{24} \cdot \pi (12)^2 = 30\pi

square meters.

Step-by-Step Solution

1

Find the radius of the circular park section using the arc length formula.

r = 12

meters

The arc length formula is

s = \frac{\theta}{360^\circ} \cdot 2\pi r

. Substituting

s = 5\pi

and

\theta = 75^\circ

gives

5\pi = \frac{75}{360} \cdot 2\pi r \implies 5\pi = \frac{5}{24} \cdot 2\pi r \implies 5\pi = \frac{5\pi r}{12} \implies r = 12

meters.

2

Calculate the sector area using the radius and central angle.

Area =

30\pi

square meters

The sector area formula is

A = \frac{\theta}{360^\circ} \cdot \pi r^2

. Substituting

\theta = 75^\circ

and

r = 12

gives

A = \frac{75}{360} \cdot \pi (12)^2 = \frac{5}{24} \cdot 144\pi = 30\pi

square meters.

Key Concept

Determining Sector Area from Central Angle and Arc Length

Question 27Question →

A rotary lawn sprinkler sweeps out a sector-shaped region of a yard with a central angle of $150^\circ$ . If the area of the irrigated sector is $60\pi$ square feet, what is the perimeter, in feet, of the irrigated lawn sector?

10\pi

10\pi + 12

10\pi + 24

5\pi + 24

20\pi + 24

Show answer & explanation

Answer: $10\pi + 24$

Answer

The perimeter of the irrigated lawn sector is

10\pi + 24

feet.

The area of a sector with central angle

150^\circ

is

\frac{150^\circ}{360^\circ} \times \pi r^2 = \frac{5}{12}\pi r^2

. Setting this equal to

60\pi

yields

r^2 = 144

, so the radius

r = 12

feet. The arc length is

\frac{5}{12} \times 2\pi(12) = 10\pi

feet. The perimeter of a sector is the arc length plus two radii (

2r

), giving

10\pi + 2(12) = 10\pi + 24

feet.

Step-by-Step Solution

1

Find the radius of the circle using the sector area formula.

r = 12

feet

The formula for sector area is

\text{Area} = \frac{\theta}{360^\circ} \cdot \pi r^2

. Substituting

\theta = 150^\circ

and

\text{Area} = 60\pi

gives

\frac{150}{360} \cdot \pi r^2 = 60\pi

, which simplifies to

\frac{5}{12} r^2 = 60

, so

r^2 = 144

and

r = 12

.

2

Calculate the arc length of the sector.

Arc length

= 10\pi

feet

The formula for arc length is

s = \frac{\theta}{360^\circ} \cdot 2\pi r

. Substituting

\theta = 150^\circ

and

r = 12

yields

s = \frac{5}{12} \cdot 2\pi (12) = 10\pi

.

3

Calculate the total perimeter of the sector.

Perimeter

= 10\pi + 24

feet

The perimeter of a sector consists of the curved arc length plus two straight radial edges:

\text{Perimeter} = s + 2r = 10\pi + 2(12) = 10\pi + 24

.

Key Concept

Perimeter of a Circle Sector

Estimated Time:1m 30s

Question 28Question →

A circular stained-glass window panel has a radius of $15$ centimeters. A specific red sector within the panel has an outer arc length of $6\pi$ centimeters. What is the area, in square centimeters, of this red sector?

15\pi

30\pi

45\pi

90\pi

225\pi

Show answer & explanation

Answer: $45\pi$

Answer

45\pi

square centimeters

The total circumference of the circle is

2\pi(15) = 30\pi

cm. The sector's arc length of

6\pi

cm is

\frac{6\pi}{30\pi} = \frac{1}{5}

of the entire circle. Since the sector area is proportional to the arc length, the area of the sector is

\frac{1}{5}

of the total area. The total area is

\pi(15)^2 = 225\pi

sq cm, so the sector area is

\frac{1}{5}(225\pi) = 45\pi

sq cm. Alternatively, using the sector area formula

A = \frac{1}{2} r s

, we get

A = \frac{1}{2}(15)(6\pi) = 45\pi

sq cm.

Step-by-Step Solution

1

Calculate the total circumference of the circle.

Circumference

C = 2\pi r = 2\pi(15) = 30\pi

cm

Knowing the total circumference allows us to determine what fraction of the circle the arc represents.

2

Determine the fractional portion of the circle corresponding to the arc.

Fraction

= \frac{\text{Arc Length}}{\text{Circumference}} = \frac{6\pi}{30\pi} = \frac{1}{5}

The central angle ratio is equivalent to the ratio of arc length to total circumference.

3

Calculate the total area of the circle.

Total Area

A_{\text{total}} = \pi r^2 = \pi (15)^2 = 225\pi

cm

^2

The sector area will be the same fraction of the total area as the arc length is of the circumference.

4

Multiply the fraction by the total area to find the sector area.

Sector Area

= \frac{1}{5} \times 225\pi = 45\pi

cm

^2

Applying the fraction

\frac{1}{5}

to the full circle area yields the targeted sector area.

Key Concept

Relationship between Arc Length and Sector Area in Circle Geometry

Question 29Question →

A wedge-shaped solar panel is constructed in the shape of a circular sector with a central angle measuring $\frac{3\pi}{4}$ radians. If the outer arc length of the panel is $9\pi$ feet, what is the area of the solar panel, in square feet?

27\pi

54\pi

81\pi

108\pi

216\pi

Show answer & explanation

Answer: $54\pi$

Answer

54\pi

square feet

First, determine the radius

r

of the sector using the arc length formula

s = r\theta

. Substituting

s = 9\pi

and

\theta = \frac{3\pi}{4}

gives

9\pi = r\left(\frac{3\pi}{4}\right)

, which simplifies to

r = 12

feet. Next, calculate the area of the sector using

A = \frac{1}{2}r^2\theta

. Substituting

r = 12

and

\theta = \frac{3\pi}{4}

yields

A = \frac{1}{2}(12)^2\left(\frac{3\pi}{4}\right) = 54\pi

square feet.

Step-by-Step Solution

1

Calculate the radius

r

of the circular sector using the arc length formula

s = r\theta

.

9\pi = r \left(\frac{3\pi}{4}\right) \implies r = 9\pi \cdot \frac{4}{3\pi} = 12\text{ feet}

The arc length

s

of a circular sector with central angle

\theta

in radians is

s = r\theta

.

2

Calculate the sector area using

A = \frac{1}{2}r^2\theta

with

r = 12

and

\theta = \frac{3\pi}{4}

.

A = \frac{1}{2}(12)^2\left(\frac{3\pi}{4}\right) = \frac{1}{2}(144)\left(\frac{3\pi}{4}\right) = 72 \cdot \frac{3\pi}{4} = 54\pi\text{ square feet}

The area of a sector with radius

r

and central angle

\theta

in radians is given by

A = \frac{1}{2}r^2\theta

.

Key Concept

Arc Length and Sector Area in Radians

Question 30Question →

A circular swimming pool cover has a total area of $64\pi$ square feet. A specific section of the cover, formed by a circular sector with a central angle of $135^\circ$ , is made of a reinforced heavy-duty material. What is the perimeter, in feet, of this reinforced sector?

6\pi

6\pi + 16

6\pi + 8

12\pi + 16

24\pi + 16

Show answer & explanation

Answer: $6\pi + 16$

Answer

The perimeter of the reinforced sector is

6\pi + 16

feet.

First, find the radius from the circle area formula:

\pi r^2 = 64\pi \implies r = 8

feet. Next, calculate the arc length of the sector using the angle fraction

\frac{135^\circ}{360^\circ} = \frac{3}{8}

. Multiplying by the total circumference

2\pi(8) = 16\pi

gives an arc length of

6\pi

feet. Finally, add the two straight radius edges to get the total perimeter:

6\pi + 8 + 8 = 6\pi + 16

feet.

Step-by-Step Solution

1

Find the radius of the circular pool cover using the given area.

Since Total Area

= \pi r^2 = 64\pi

, dividing by

\pi

yields

r^2 = 64

, so

r = 8

feet.

The radius is required to determine both the arc length and the lengths of the straight boundary edges.

2

Calculate the arc length of the sector.

Arc length

= \frac{\theta}{360^\circ} \times 2\pi r = \frac{135^\circ}{360^\circ} \times 2\pi(8) = \frac{3}{8} \times 16\pi = 6\pi

feet.

The curved boundary of the sector is a fraction of the circle's full circumference.

3

Calculate the total perimeter of the sector by adding the arc length and the two radii.

Perimeter

= \text{Arc length} + 2r = 6\pi + 2(8) = 6\pi + 16

feet.

The perimeter of any sector consists of its curved outer arc plus its two straight side radii.

Key Concept

Perimeter of a Circular Sector

Estimated Time:1m 0s

Question 31Question →

A lighthouse beacon sweeps across a circular region of the ocean. The illuminated area forms a circular sector with a radius of $15$ nautical miles and an area of $75\pi$ square nautical miles. What is the total perimeter, in nautical miles, of this illuminated sector?

10\pi

10\pi + 15

10\pi + 30

20\pi + 30

5\pi + 30

Show answer & explanation

Answer: $10\pi + 30$

Answer

10\pi + 30

nautical miles

The full circle has an area of

\pi(15)^2 = 225\pi

. The illuminated sector's area of

75\pi

is

\frac{75\pi}{225\pi} = \frac{1}{3}

of the total circle. Thus, its arc length is

\frac{1}{3}

of the total circumference

2\pi(15) = 30\pi

, giving an arc length of

10\pi

. Adding the two bounding radii of length

15

each yields a total sector perimeter of

10\pi + 30

nautical miles.

Step-by-Step Solution

1

Calculate the total area of the full circle.

Total Area =

\pi r^2 = \pi (15)^2 = 225\pi

square nautical miles.

Knowing the full circle area allows us to find what fraction of the circle the sector occupies.

2

Determine the fractional proportion of the sector relative to the whole circle.

Fraction =

\frac{\text{Sector Area}}{\text{Total Area}} = \frac{75\pi}{225\pi} = \frac{1}{3}

.

The sector represents

\frac{1}{3}

of both the total area and the total circumference of the circle.

3

Calculate the arc length of the sector.

Arc length =

\frac{1}{3} \times 2\pi r = \frac{1}{3} \times 30\pi = 10\pi

nautical miles.

The curved boundary of the sector is

\frac{1}{3}

of the circle's full circumference.

4

Calculate the total perimeter of the sector by adding the arc length and the two straight radii.

Perimeter = Arc length

+ 2r = 10\pi + 15 + 15 = 10\pi + 30

nautical miles.

The perimeter of a circular sector consists of its curved outer arc plus its two straight bounding radii.

Key Concept

Perimeter of a Circular Sector

Question 32Question →

A circular theater stage has a designated performance section shaped as a circular sector with a central angle measuring $\frac{2\pi}{5}$ radians. If the arc length along the outer edge of this sector is $8\pi$ feet, what is the area, in square feet, of the performance section?

20\pi

40\pi

80\pi

160\pi

400\pi

Show answer & explanation

Answer: $80\pi$

Answer

The area of the performance sector is

80\pi

square feet.

First, determine the radius of the circle using the radian arc length formula

s = r\theta

. Substituting

s = 8\pi

and

\theta = \frac{2\pi}{5}

yields

r = 20

feet. Next, calculate the area of the circular sector using

A = \frac{1}{2}r^2\theta = \frac{1}{2}(20)^2\left(\frac{2\pi}{5}\right) = 80\pi

square feet (or equivalently

A = \frac{1}{2}rs = \frac{1}{2}(20)(8\pi) = 80\pi

square feet).

Step-by-Step Solution

1

Use the arc length formula in radians to solve for the radius (

r

).

Since

s = r\theta

, substituting

s = 8\pi

and

\theta = \frac{2\pi}{5}

gives

8\pi = r \cdot \frac{2\pi}{5}

, so

r = 8\pi \cdot \frac{5}{2\pi} = 20

feet.

The radius of the circular stage is required to calculate the sector area.

2

Calculate the area of the sector using the formula

A = \frac{1}{2}r^2\theta

(or

A = \frac{1}{2}rs

).

A = \frac{1}{2}(20)^2\left(\frac{2\pi}{5}\right) = \frac{1}{2}(400)\left(\frac{2\pi}{5}\right) = 200\left(\frac{2\pi}{5}\right) = 80\pi

square feet.

Applying the known radius and central angle into the sector area formula yields the final area.

Key Concept

Calculating sector area from arc length and central angle in radians

Estimated Time:1m 15s

Question 33Question →

A circular radar display at an air traffic control tower has a radius of $12$ inches. A wedge-shaped tracking zone on the display is bounded by two radii and an outer arc length of $5\pi$ inches. What is the area, in square inches, of this tracking zone?

15\pi

30\pi

60\pi

120\pi

144\pi

Show answer & explanation

Answer: $30\pi$

Answer

The area of the tracking zone is

30\pi

square inches.

The total area of the circle is

\pi(12)^2 = 144\pi

square inches, and the total circumference is

2\pi(12) = 24\pi

inches. The tracking sector accounts for a fraction of

\frac{5\pi}{24\pi} = \frac{5}{24}

of the full circle. Multiplying this fraction by the total area yields

\frac{5}{24} \times 144\pi = 30\pi

square inches.

Step-by-Step Solution

1

Calculate the total circumference and total area of the circular display

Circumference

C = 2\pi r = 2\pi(12) = 24\pi

inches, and Total Area

A = \pi r^2 = \pi(12)^2 = 144\pi

square inches.

These total values establish the scale needed to find the fractional sector area.

2

Find the fraction of the circle represented by the outer arc length

Fraction

= \frac{\text{Arc Length}}{\text{Circumference}} = \frac{5\pi}{24\pi} = \frac{5}{24}

.

The ratio of an arc length to the total circumference equals the ratio of the sector area to the total area.

3

Multiply the fraction by the total area of the circle

\text{Sector Area} = \frac{5}{24} \times 144\pi = 5 \times 6\pi = 30\pi$ square inches.

Applying the arc length fraction to the total area yields the exact sector area.

Key Concept

Sector Area and Arc Length Proportions

Estimated Time:1m 0s

Question 34Question →

A circular grinding wheel has a radius of $9$ inches. A section of the wheel bounded by a central angle of $140^\circ$ is coated with a coarse abrasive layer. What is the area, in square inches, of the remaining non-coated section of the grinding wheel?

11\pi

31.5\pi

49.5\pi

63\pi

99\pi

Show answer & explanation

Answer: $49.5\pi$

Answer

The area of the non-coated section of the grinding wheel is

49.5\pi

square inches.

The central angle of the entire circle is

360^\circ

. Since the coated portion occupies

140^\circ

, the non-coated portion occupies

360^\circ - 140^\circ = 220^\circ

. The total area of the circle is

\pi (9)^2 = 81\pi

. Multiplying the total area by the fraction

\frac{220^\circ}{360^\circ}

yields

\frac{11}{18} \cdot 81\pi = 49.5\pi

square inches.

Step-by-Step Solution

1

Find the central angle of the non-coated section

360^\circ - 140^\circ = 220^\circ

A full circle measures

360^\circ

, so subtracting the coated angle gives the non-coated angle.

2

Calculate the total area of the circular grinding wheel

\text{Area}_{\text{total}} = \pi r^2 = \pi (9^2) = 81\pi\text{ sq in}

The total area formula for a circle with radius

r

is

\pi r^2

.

3

Calculate the area of the non-coated sector

\text{Area}_{\text{sector}} = \frac{220^\circ}{360^\circ} \times 81\pi = \frac{11}{18} \times 81\pi = 49.5\pi\text{ sq in}

Multiply the total area by the fraction of the circle formed by the central angle.

Key Concept

Sector Area of a Circle

Question 35Question →

A circular tabletop has a radius of $30$ inches. A wooden section shaped as a circular sector covers a portion of the tabletop defined by a central angle of $108^\circ$ . What is the area, in square inches, of this wooden sector section?

9\pi

18\pi

270\pi

540\pi

1080\pi

Show answer & explanation

Answer: $270\pi$

Answer

The area of the wooden sector section is

270\pi

square inches.

The area of a sector of a circle with radius

r

and central angle

\theta

(in degrees) is given by the formula

A = \frac{\theta}{360^\circ} \cdot \pi r^2

. Substituting

r = 30

inches and

\theta = 108^\circ

yields

A = \frac{108}{360} \cdot \pi (30)^2 = \frac{3}{10} \cdot 900\pi = 270\pi

square inches.

Step-by-Step Solution

1

Calculate the total area of the circular tabletop.

Total Area =

\pi r^2 = \pi (30)^2 = 900\pi

square inches.

The area of a full circle with radius

r

is given by

A = \pi r^2

.

2

Determine the fraction of the circle represented by the central angle.

Fraction =

\frac{108^\circ}{360^\circ} = \frac{3}{10}

.

A full circle measures

360^\circ

, so the central angle forms a fraction

\frac{\theta}{360^\circ}

of the total circle.

3

Multiply the fraction by the total area of the circle to find the sector area.

Sector Area =

\frac{3}{10} \cdot 900\pi = 270\pi

square inches.

The sector area is proportional to the fraction of the central angle relative to the full circle.

Key Concept

Sector Area Formula

Question 36Question →

A circular metal disk has a radius of $12$ centimeters. A wedge-shaped sector with a central angle of $75^\circ$ is cut out and removed from the disk. What is the perimeter, in centimeters, of the remaining portion of the disk?

5\pi

5\pi + 24

19\pi

19\pi + 12

19\pi + 24

Show answer & explanation

Answer: $19\pi + 24$

Answer

19\pi + 24

The central angle of the remaining portion is

360^\circ - 75^\circ = 285^\circ

. The arc length of this remaining sector is

\frac{285^\circ}{360^\circ} \times 2\pi(12) = 19\pi

cm. Because cutting out the wedge exposes two straight sides equal to the radius (

12

cm each), the total perimeter is the sum of the curved arc and the two radii:

19\pi + 12 + 12 = 19\pi + 24

cm.

Step-by-Step Solution

1

Find the central angle of the remaining major sector

360^\circ - 75^\circ = 285^\circ

Removing a

75^\circ

wedge from a full

360^\circ

circle leaves a central angle of

285^\circ

.

2

Calculate the arc length of the remaining major sector

Arc length

= \frac{285^\circ}{360^\circ} \times 2\pi(12) = \frac{19}{24} \times 24\pi = 19\pi

cm

Arc length is given by the formula

s = \frac{\theta}{360^\circ} \times 2\pi r

.

3

Calculate the total perimeter of the remaining shape

Perimeter

= 19\pi + 12 + 12 = 19\pi + 24

cm

The total boundary of the remaining piece consists of the curved arc plus the two straight straight edges formed by radii where the cut occurred.

Key Concept

Perimeter of a Sector and Arc Length Formula

Estimated Time:1m 0s

Question 37Question →

A section of a circular garden is enclosed by two radii and an outer arc, forming a circular sector with a central angle of $60^\circ$ . If the total perimeter of this sector is $12 + 2\pi$ meters, what is the area of the sector, in square meters?

2\pi

3\pi

6\pi

12\pi

36\pi

Show answer & explanation

Answer: $6\pi$

Answer

The area of the sector is

6\pi

square meters.

The perimeter of a circular sector is the sum of its two straight edges (radii) and its curved edge (arc length):

\text{Perimeter} = 2r + \frac{\theta}{360^\circ} \cdot 2\pi r

. Substituting

\theta = 60^\circ

gives

2r + \frac{\pi r}{3} = 12 + 2\pi

. Matching corresponding terms yields

2r = 12

, so

r = 6

meters. The sector area is then

\frac{60^\circ}{360^\circ} \cdot \pi (6)^2 = 6\pi

square meters.

Step-by-Step Solution

1

Express the arc length

s

in terms of the radius

r

.

s = \frac{60^\circ}{360^\circ} \cdot 2\pi r = \frac{1}{6} \cdot 2\pi r = \frac{\pi r}{3}

meters.

The arc length of a sector with central angle

\theta

in degrees is given by

s = \frac{\theta}{360^\circ} \cdot 2\pi r

.

2

Set up and solve the equation for the sector's total perimeter to find radius

r

.

Perimeter

= 2r + s = 2r + \frac{\pi r}{3} = 12 + 2\pi

. Equating integer and

\pi

components gives

2r = 12 \implies r = 6

meters.

The perimeter of a sector consists of the two bounding radii plus the arc length.

3

Calculate the area of the sector using

r = 6

meters.

Sector Area

= \frac{60^\circ}{360^\circ} \cdot \pi r^2 = \frac{1}{6} \cdot \pi (6)^2 = \frac{36\pi}{6} = 6\pi

square meters.

The area of a circular sector is given by

A = \frac{\theta}{360^\circ} \cdot \pi r^2

.

Key Concept

Perimeter and Area of a Circular Sector

Estimated Time:1m 30s

Question 38Question →

A jeweler is creating a circular gold pendant with a radius of $18$ millimeters. A section of the pendant is shaped as a circular sector and has an arc length of $15\pi$ millimeters. What is the area, in square millimeters, of this sector?

30\pi

108\pi

135\pi

189\pi

270\pi

Show answer & explanation

Answer: $135\pi$

Answer

The area of the sector is

135\pi

square millimeters.

The sector area is calculated by applying the proportion of the arc length to the full circumference:

\text{Sector Area} = \frac{\text{Arc Length}}{2\pi r} \times \pi r^2 = \frac{1}{2} r s

. Substituting

r = 18

mm and

s = 15\pi

mm gives

\frac{1}{2} (18)(15\pi) = 135\pi

square millimeters.

Step-by-Step Solution

1

Find the ratio of the central angle to the full circle using arc length

The circumference of the circle is

C = 2\pi r = 2\pi (18) = 36\pi

mm. The sector's arc length fraction is

\frac{15\pi}{36\pi} = \frac{5}{12}

.

Arc length is proportional to the total circumference of the circle.

2

Calculate the total area of the circle

\text{Total Area} = \pi r^2 = \pi (18)^2 = 324\pi

square millimeters.

The total area formula for a circle of radius

r

is

\pi r^2

.

3

Multiply the total area by the sector fraction to find the sector area

\text{Sector Area} = \frac{5}{12} \times 324\pi = 135\pi$ square millimeters.

The area of a circular sector is the same fraction of the total area as its arc length is of the circumference.

Key Concept

Relationship between arc length, radius, and sector area in a circle

Estimated Time:1m 0s

Question 39Question →

A board game spinner has a pointer of length $6\text{ inches}$ that pivots at the center of the circular board. If the sector of the board representing 'Lose a Turn' has a central angle of $\frac{\pi}{3}$ radians, what is the area, in square inches, of this sector?

2\pi

\frac{3\pi}{2}

6\pi

12\pi

36\pi

Show answer & explanation

Answer: $6\pi$

Answer

6\pi

square inches

The area of a sector with a radius

r

and a central angle

\theta

(measured in radians) is determined by the formula

A = \frac{1}{2}r^2\theta

. Given that the radius

r = 6

and the central angle

\theta = \frac{\pi}{3}

, substituting these values into the formula gives

A = \frac{1}{2}(6)^2\left(\frac{\pi}{3}\right) = \frac{1}{2}(36)\left(\frac{\pi}{3}\right) = 6\pi

square inches.

Step-by-Step Solution

1

Identify the radius and the central angle of the sector from the problem description.

Radius

r = 6\text{ inches}

, and central angle

\theta = \frac{\pi}{3}\text{ radians}

.

These values are the direct inputs required for the circle sector formulas.

2

Use the sector area formula in radians,

A = \frac{1}{2}r^2\theta

, to calculate the area.

A = \frac{1}{2}(6)^2\left(\frac{\pi}{3}\right) = \frac{1}{2}(36)\left(\frac{\pi}{3}\right) = 18\left(\frac{\pi}{3}\right) = 6\pi

.

This formula scales the total area of the circle by the fraction represented by the radian angle relative to

2\pi

radians.

Key Concept

Calculating the area of a circle sector using radian measure

Estimated Time:1m 0s

Question 40Question →

A coastal lighthouse beacon rotates through a central angle of $135^\circ$ , sweeping across a circular sector of sea with a radius of $12\text{ nautical miles}$ . What is the total perimeter, in nautical miles, of the region swept by the beacon?

9\pi

9\pi + 12

9\pi + 24

54\pi

54\pi + 24

Show answer & explanation

Answer: $9\pi + 24$

Answer

The total perimeter of the swept sector is

9\pi + 24

nautical miles.

The perimeter of a sector consists of the curved arc length plus the lengths of the two straight radii bounding the sector. The arc length is calculated as

\frac{135}{360} \times 2\pi(12) = \frac{3}{8} \times 24\pi = 9\pi

. Adding the two radii of

12

nautical miles each gives

9\pi + 12 + 12 = 9\pi + 24

nautical miles.

Step-by-Step Solution

1

Calculate the arc length of the sector

Arc length =

\frac{135^\circ}{360^\circ} \times 2\pi(12) = \frac{3}{8} \times 24\pi = 9\pi

nautical miles.

The curved outer boundary of a circular sector is determined by the fraction of the total circumference defined by the central angle.

2

Determine the total perimeter of the sector

Perimeter =

\text{Arc length} + 2r = 9\pi + 2(12) = 9\pi + 24

nautical miles.

The total boundary of a sector consists of its outer curved arc plus its two straight radial line segments.

Key Concept

Perimeter of a Circular Sector

Estimated Time:1m 30s

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