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

Question 1Question →

A circular pizza with a radius of $8\text{ inches}$ is cut into slices. If one slice has a central angle of $45^\circ$ , what is the area, in square inches, of this slice?

2\pi

8\pi

16\pi

56\pi

64\pi

Show answer & explanation

Answer: $8\pi$

Answer

The area of the slice is

8\pi

square inches.

The area of a sector is found by multiplying the circle's total area by the fraction of the circle that the sector represents. With a radius of

8

inches, the total area is

\pi \times 8^2 = 64\pi

square inches. Since the slice has a central angle of

45^\circ

, it represents

\frac{45}{360} = \frac{1}{8}

of the entire pizza. Multiplying the total area by this fraction gives

\frac{1}{8} \times 64\pi = 8\pi

square inches.

Step-by-Step Solution

1

Calculate the total area of the circular pizza using the area formula

A = \pi r^2

with

r = 8

.

The total area of the pizza is

\pi \times 8^2 = 64\pi

square inches.

To find the area of a sector, we first need to determine the area of the entire circle.

2

Find the fraction of the circle represented by the slice's central angle by dividing

45^\circ

by

360^\circ

.

The fraction is

\frac{45}{360} = \frac{1}{8}

.

A full circle has

360^\circ

, so the ratio of the central angle to

360^\circ

gives the proportion of the circle's total area that the sector occupies.

3

Multiply the total area of the pizza by the fraction representing the slice.

The area of the slice is

\frac{1}{8} \times 64\pi = 8\pi

square inches.

Multiplying the total circle area by the sector's fraction yields the sector's area.

Key Concept

The area of a sector is proportional to its central angle and can be found using the formula

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

when the angle is in degrees.

Question 2Question →

A circular metal plate has a radius of $6\text{ inches}$ . A sector of the plate with a central angle of $60^\circ$ is cut out to make a custom spacer. What is the area, in square inches, of the cut-out sector?

2\pi

\frac{3\pi}{2}

6\pi

\frac{36\pi}{5}

36\pi

Show answer & explanation

Answer: $6\pi$

Answer

The correct area of the sector is

6\pi

square inches.

The correct answer of

6\pi

is found by calculating the total area of the circular plate using

A = \pi r^2 = \pi (6)^2 = 36\pi

, and then multiplying this by the ratio of the central angle to the total degrees in a circle, which is

\frac{60^\circ}{360^\circ} = \frac{1}{6}

. Thus, the sector area is

\frac{1}{6} \times 36\pi = 6\pi

square inches.

Step-by-Step Solution

1

Calculate the area of the entire circular plate.

The total area of the circle is

A = \pi (6)^2 = 36\pi

square inches.

Before finding the area of a sector, we need the total area of the circle of which it is a part.

2

Determine the fraction of the circle represented by the sector.

The fraction is

\frac{60^\circ}{360^\circ} = \frac{1}{6}

.

A circle contains

360^\circ

, so a central angle of

60^\circ

corresponds to

\frac{60}{360}

of the full circle.

3

Multiply the total area of the circle by the fraction.

The sector area is

36\pi \times \frac{1}{6} = 6\pi

square inches.

The area of a sector is proportional to its central angle relative to the total angle of a circle.

Key Concept

The area of a sector with radius

r

and central angle

\theta

in degrees is given by the formula

A = \pi r^2 \left(\frac{\theta}{360^\circ}\right)

.

Alternative Method

Alternatively, since

60^\circ

is

\frac{1}{6}

of a full

360^\circ

circle, the sector area is simply one-sixth of the total area of the circle (

36\pi

), which gives

6\pi

square inches.

Estimated Time:45s

Question 3Question →

A pendulum of a certain length swings back and forth such that the tip of the pendulum traces an arc of length $6\pi$ inches. If the length of the pendulum is increased by $4$ inches and it swings through the same central angle, the tip of the pendulum traces an arc of length $8\pi$ inches. What is the area, in square inches, of the circular sector swept out by the original pendulum?

12\pi

18\pi

36\pi

72\pi

144\pi

Show answer & explanation

Answer: $36\pi$

Answer

The correct answer is

36\pi

square inches.

The correct answer is

36\pi

square inches. By using the arc length formula

s = r\theta

where

\theta

is in radians, we establish the system of equations

r\theta = 6\pi

and

(r+4)\theta = 8\pi

. Subtracting the first equation from the second gives

4\theta = 2\pi

, which simplifies to

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

. Substituting this back into the first equation yields

r(\frac{\pi}{2}) = 6\pi

, so the original radius

r = 12

. Finally, we calculate the area of the original sector using the formula

A = \frac{1}{2}r^2\theta = \frac{1}{2}(12)^2(\frac{\pi}{2}) = 36\pi

.

Step-by-Step Solution

1

Write down the arc length equations for both pendulums using the formula

s = r\theta

, where

r

is the length of the pendulum (radius) and

\theta

is the central angle in radians.

For the original pendulum:

r\theta = 6\pi

. For the extended pendulum:

(r+4)\theta = 8\pi

.

To establish the mathematical relationships between the given arc lengths, the pendulum lengths, and the central angle.

2

Solve the system of equations for the central angle

\theta

.

Distributing the second equation gives

r\theta + 4\theta = 8\pi

. Substituting

r\theta = 6\pi

into this yields

6\pi + 4\theta = 8\pi \implies 4\theta = 2\pi \implies \theta = \frac{\pi}{2}

radians.

To find the constant central angle of the pendulum's swing.

3

Substitute

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

back into the first equation to solve for the original radius

r

.

r\left(\frac{\pi}{2}\right) = 6\pi \implies r = 12

inches.

To find the length of the original pendulum, which serves as the radius of the sector.

4

Calculate the area of the sector swept out by the original pendulum using the formula

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

.

A = \frac{1}{2}(12)^2\left(\frac{\pi}{2}\right) = \frac{1}{2}(144)\left(\frac{\pi}{2}\right) = 36\pi

square inches.

To find the final area of the sector as requested by the question.

Key Concept

Calculating sector area using arc length relationships to determine radius and angle.

Alternative Method

Instead of solving for

\theta

first, one can note that the ratio of the arc lengths is equal to the ratio of the radii because the central angle is constant:

\frac{r}{r+4} = \frac{6\pi}{8\pi} = \frac{3}{4}

. Solving for

r

gives

4r = 3r + 12 \implies r = 12

. Since the arc length of the original sector is

s = 6\pi

, we can use the sector area formula

A = \frac{1}{2}rs = \frac{1}{2}(12)(6\pi) = 36\pi

square inches.

Estimated Time:3m 0s

Question 4Question →

A windshield wiper of length $15\text{ inches}$ sweeps through a central angle of $120^\circ$ across a windshield. What is the area, in square inches, of the region swept by the wiper?

10\pi

75\pi

112.5\pi

150\pi

225\pi

Show answer & explanation

Answer: $75\pi$

Answer

75\pi

square inches

The area of the region swept by the wiper is the area of a circle sector with a radius of

15\text{ inches}

and a central angle of

120^\circ

. The formula for the area of a sector is

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

. Substituting

15

for the radius and

120

for the angle gives

A = \frac{120}{360} \times \pi (15)^2 = \frac{1}{3} \times 225\pi = 75\pi

square inches.

Step-by-Step Solution

1

Determine the formula for the area of a sector of a circle.

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

The area of a sector is a fraction of the total area of the circle, where the fraction is determined by the central angle

\theta

divided by the total degrees in a circle (

360^\circ

).

2

Substitute the given values into the sector area formula, using a radius of

15

and a central angle of

120^\circ

.

A = \frac{120^\circ}{360^\circ} \times \pi (15)^2

The windshield wiper length represents the radius

r = 15\text{ inches}

, and the sweep angle represents the central angle

\theta = 120^\circ

.

3

Simplify the expression to find the final area.

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

Reducing the fraction

\frac{120}{360}

to

\frac{1}{3}

and squaring

15

to get

225

yields the area of

75\pi

square inches.

Key Concept

The area of a circle sector is found by multiplying the total circle area,

\pi r^2

, by the ratio of the central angle to the total degree measure of a circle,

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

.

Question 5Question →

A circle is inscribed inside a sector of a larger circle. The larger circle has a radius of $18\text{ inches}$ and the sector has a central angle of $60^\circ$ , as shown in the figure. What is the area, in square inches, of the region that is inside the sector but outside the inscribed circle?

27\pi

45\pi

18\pi

72\pi

288\pi

Show answer & explanation

Answer: $18\pi$

Answer

The correct area is

18\pi

square inches.

The correct answer is

18\pi

square inches. First, the sector area is calculated as

\frac{60}{360} \times \pi \times 18^2 = 54\pi

. Second, using the right triangle formed by the sector's center, the inscribed circle's center, and the point of tangency, we set up

\sin(30^\circ) = \frac{r}{18-r}

. Solving for

r

gives

r = 6

, so the area of the inscribed circle is

\pi \times 6^2 = 36\pi

. The difference between the two areas is

54\pi - 36\pi = 18\pi

.

Step-by-Step Solution

1

Calculate the area of the

60^\circ

sector of the larger circle.

Sector Area =

54\pi

square inches

The sector has a radius of

R = 18

and a central angle of

60^\circ

. The sector's area is a fraction of the total circle's area:

\text{Area}_{\text{sector}} = \frac{60}{360} \times \pi R^2 = \frac{1}{6} \times \pi \times 18^2 = \frac{324\pi}{6} = 54\pi

.

2

Determine the radius

r

of the inscribed circle using trigonometry.

Inscribed Radius

r = 6

inches

The center of the inscribed circle,

I

, lies on the angle bisector of the sector. The line segment from the center of the sector

O

to

I

bisects the

60^\circ

angle, forming a

30^\circ

angle. The distance from

O

to the outer boundary of the sector is

R = 18

, and the distance from

I

to the boundary is

r

, so the hypotenuse

OI = 18 - r

. Drawing a perpendicular from

I

to one of the straight edges of the sector creates a right triangle with opposite side

r

(the radius) and hypotenuse

18 - r

. Applying the sine ratio:

\sin(30^\circ) = \frac{r}{18 - r}

. Since

\sin(30^\circ) = 0.5

, we solve

0.5 = \frac{r}{18 - r} \Rightarrow 18 - r = 2r \Rightarrow 3r = 18 \Rightarrow r = 6

.

3

Calculate the area of the inscribed circle.

Inscribed Circle Area =

36\pi

square inches

The area of the inscribed circle with radius

r = 6

is

\text{Area}_{\text{circle}} = \pi r^2 = \pi \times 6^2 = 36\pi

.

4

Subtract the area of the inscribed circle from the area of the sector.

Remaining Area =

18\pi

square inches

The area of the region inside the sector but outside the circle is found by subtracting the inscribed circle area from the sector area:

54\pi - 36\pi = 18\pi

.

Key Concept

Using trigonometry on angle bisectors to determine the radius of a circle inscribed inside a sector, and applying sector and circle area formulas.

Estimated Time:3m 0s

Question 6Question →

A dog is tied to a post in the center of a flat, grassy yard with a leash that is $12\text{ feet}$ long. If the dog walks along an arc formed by a central angle of $\frac{\pi}{3}$ radians, what is the length, in feet, of the path the dog travels?

2\pi

4\pi

8\pi

12\pi

24\pi

Show answer & explanation

Answer: $4\pi$

Answer

The correct option is the one stating

4\pi

.

The length of the path traveled by the dog is the arc length of a circle with a radius of

12

feet and a central angle of

\frac{\pi}{3}

radians. Using the radian arc length formula,

s = r\theta

, substituting

r = 12

and

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

yields

s = 12 \times \frac{\pi}{3} = 4\pi

feet.

Step-by-Step Solution

1

Identify the given values from the problem statement.

The radius of the circular path is

r = 12

feet, and the central angle is

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

radians.

These parameters are required to calculate the arc length of the path.

2

Recall the formula for the arc length of a circle when the angle is in radians.

The formula is

s = r\theta

.

Since the angle is given in radians, the arc length is directly the product of the radius and the angle.

3

Substitute the values into the formula and calculate the result.

s = 12 \times \frac{\pi}{3} = 4\pi

.

Multiplying the radius by the angle in radians gives the length of the path traveled.

Key Concept

Calculating the arc length of a circle when the central angle is measured in radians using the formula

s = r\theta

.

Question 7Question →

A goat is tethered to one of the outer corners of a flat, rectangular shed that measures $6\text{ meters}$ by $8\text{ meters}$ . The tether is $10\text{ meters}$ long. Assuming the goat remains outside the shed, what is the total area, in square meters, of the region the goat can graze?

75\pi

83\pi

80\pi

90\pi

100\pi

Show answer & explanation

Answer: $80\pi$

Answer

The correct grazing area is

80\pi

square meters.

The total grazing area is the sum of three sectors: a sector of radius

10\text{ meters}

with central angle

270^\circ

(area

75\pi

), a sector of radius

4\text{ meters}

with central angle

90^\circ

(area

4\pi

), and a sector of radius

2\text{ meters}

with central angle

90^\circ

(area

\pi

). Adding these yields

80\pi

square meters.

Step-by-Step Solution

1

Determine the area of the main sector.

The main sector has a radius of

10\text{ meters}

and a central angle of

360^\circ - 90^\circ = 270^\circ

. The area is

\frac{270}{360} \times \pi \times 10^2 = 75\pi

square meters.

The corner of the rectangular shed blocks

90^\circ

of a full circle, leaving a

270^\circ

sector.

2

Determine the area of the sector at the corner adjacent to the

6

-meter side.

The tether wraps around the corner, leaving a remaining length of

10 - 6 = 4\text{ meters}

. It sweeps through an angle of

90^\circ

. The area is

\frac{90}{360} \times \pi \times 4^2 = 4\pi

square meters.

When the tether extends past the adjacent corner along the

6

-meter side, the pivot point becomes that corner and the tether length decreases by the side length.

3

Determine the area of the sector at the corner adjacent to the

8

-meter side.

The tether wraps around the corner, leaving a remaining length of

10 - 8 = 2\text{ meters}

. It sweeps through an angle of

90^\circ

. The area is

\frac{90}{360} \times \pi \times 2^2 = \pi

square meters.

When the tether extends past the adjacent corner along the

8

-meter side, the pivot point becomes that corner and the tether length decreases by the side length.

4

Check for overlap and calculate the total grazing area.

The total area is

75\pi + 4\pi + \pi = 80\pi

square meters.

The two smaller sectors are at different corners of the rectangular shed and do not overlap. Summing the three sector areas gives the total grazing region.

Key Concept

Calculating sector areas by determining the correct radii and central angles based on geometric constraints.

Question 8Question →

A circle has a radius of $12\text{ centimeters}$ . What is the length, in centimeters, of the arc intercepted by a central angle of $30^\circ$ ?

\pi

2\pi

12\pi

24\pi

2

Show answer & explanation

Answer: $2\pi$

Answer

2\pi

centimeters

To find the arc length, multiply the total circumference of the circle,

2\pi r

, by the fraction of the circle represented by the central angle,

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

. With a radius of

12

centimeters and a central angle of

30^\circ

, this calculation yields

2\pi (12) \times \frac{30}{360} = 24\pi \times \frac{1}{12} = 2\pi

centimeters.

Step-by-Step Solution

1

Identify the formula for arc length:

s = 2\pi r \left(\frac{\theta}{360^\circ}\right)

, where

r

is the radius and

\theta

is the central angle in degrees.

Formula established:

s = 2\pi r \left(\frac{\theta}{360^\circ}\right)

Arc length is the fraction of the total circumference determined by the central angle.

2

Substitute the given radius

r = 12\text{ cm}

and central angle

\theta = 30^\circ

into the formula.

Equation set up:

s = 2\pi (12) \left(\frac{30}{360}\right)

This sets up the specific calculation for the given circle.

3

Simplify the expression to find the final arc length.

s = 24\pi \left(\frac{1}{12}\right) = 2\pi

Simplification yields the exact length of the arc in terms of

\pi

.

Key Concept

The length of an arc is proportional to the fraction of the circle's circumference represented by the central angle.

Estimated Time:1m 0s

Question 9Question →

A circular garden plot has a radius of $10\text{ meters}$ . A sector of the garden with a central angle of $72^\circ$ is planted with roses. What is the area, in square meters, of the sector planted with roses?

4\pi

40\pi

20\pi

72\pi

100\pi

Show answer & explanation

Answer: $20\pi$

Answer

The correct answer is

20\pi

, representing the area of the sector in square meters.

The area of a sector of a circle is calculated using the formula

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

. Substituting

\theta = 72^\circ

and

r = 10\text{ meters}

gives

\frac{72^\circ}{360^\circ} \pi (10)^2 = \frac{1}{5} \cdot 100\pi = 20\pi

.

Step-by-Step Solution

1

Identify the formula for the area of a sector.

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

The area of a sector is a proportional fraction of the total area of the circle.

2

Substitute the given values into the formula.

\text{Area} = \frac{72^\circ}{360^\circ} \pi (10)^2

The central angle is

72^\circ

and the radius is

10\text{ meters}

.

3

Simplify the fraction and the squared term.

\text{Area} = \frac{1}{5} \pi (100)

72/360

simplifies to

1/5

and

10^2 = 100

.

4

Perform the final multiplication.

20\pi

One-fifth of

100

is

20

.

Key Concept

Calculating the area of a circle sector given the radius and the central angle in degrees.

Estimated Time:1m 0s

Question 10Question →

A circular archery target has a radius of $12\text{ inches}$ . A sector of this target has a central angle of $150^\circ$ . What is the area, in square inches, of this sector?

10\pi

15\pi

60\pi

120\pi

240\pi

Show answer & explanation

Answer: $60\pi$

Answer

The correct area of the sector is

60\pi

square inches.

To find the area of a sector, first calculate the total area of the circle, which is

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

square inches. Then, multiply this total area by the fraction of the circle that the sector represents:

\frac{150^\circ}{360^\circ} = \frac{5}{12}

. Calculating the product gives

\frac{5}{12} \times 144\pi = 60\pi

square inches.

Step-by-Step Solution

1

Find the total area of the circular archery target using the area formula

A = \pi r^2

with a radius of

12\text{ inches}

.

The total area of the circle is

\pi (12)^2 = 144\pi

square inches.

The area of a sector is a fractional part of the circle's total area.

2

Calculate the fraction of the circle represented by a central angle of

150^\circ

.

The fraction is

\frac{150^\circ}{360^\circ} = \frac{5}{12}

.

A complete circle has a central angle of

360^\circ

.

3

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

The sector area is

\frac{5}{12} \times 144\pi = 60\pi

square inches.

Applying the fraction to the total area yields the area of the sector.

Key Concept

Calculating the area of a circle sector using the formula

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

.

Estimated Time:1m 0s

Question 11Question →

A circular sector with radius $R$ and central angle $\theta$ (measured in radians) has the same area and the same perimeter as a square with side length $s$ . What is the value of $\theta$ ?

2

4

8

\frac{1}{\pi}

2\pi

Show answer & explanation

Answer: 2

Answer

The value of theta must be 2

The correct value is 2. By equating the sector's area and perimeter to the square's area and perimeter, we set up a system of equations:

\frac{1}{2} R^2 \theta = s^2

and

2R + R\theta = 4s

. Solving for

s

from the area equation gives

s = R\sqrt{\theta/2}

. Substituting this into the perimeter equation and dividing by

R

yields

2 + \theta = 2\sqrt{2\theta}

. Squaring both sides and simplifying leads to the quadratic equation

\theta^2 - 4\theta + 4 = 0

, which has a single solution of

\theta = 2

radians.

Step-by-Step Solution

1

Write down the equations for the area and perimeter of both shapes.

For the circular sector: Area =

\frac{1}{2}R^2\theta

, Perimeter =

2R + R\theta

(where

\theta

is in radians). For the square: Area =

s^2

, Perimeter =

4s

.

These are the standard geometric formulas for a sector in radians and a square.

2

Set the corresponding areas and perimeters equal to each other to form a system of equations.

Equation 1:

\frac{1}{2} R^2 \theta = s^2

Equation 2:

2R + R\theta = 4s

The problem statement specifies that the two shapes have equal areas and equal perimeters.

3

Solve Equation 1 for the side length

s

of the square.

s = R \sqrt{\frac{\theta}{2}}

This allows us to substitute

s

in Equation 2 and solve for

\theta

in terms of

R

.

4

Substitute the expression for

s

into Equation 2 and simplify.

2R + R\theta = 4\left(R \sqrt{\frac{\theta}{2}}\right) \implies 2 + \theta = 4\sqrt{\frac{\theta}{2}}

Since the radius

R

is a positive length, we can divide both sides of the equation by

R

.

5

Solve the simplified equation for

\theta

by squaring both sides.

2 + \theta = 2\sqrt{2\theta} \implies (2+\theta)^2 = (2\sqrt{2\theta})^2 \implies 4 + 4\theta + \theta^2 = 8\theta \implies \theta^2 - 4\theta + 4 = 0 \implies (\theta - 2)^2 = 0 \implies \theta = 2

Squaring both sides and setting the quadratic equation to zero yields a perfect square trinomial with a single real solution.

Key Concept

Relating the area and perimeter of a circular sector using radian measures to those of a square.

Estimated Time:3m 0s

Question 12Question →

A sector of a circle has a central angle of $\frac{3\pi}{4}$ radians and an area of $3\pi$ square inches. What is the perimeter, in inches, of the sector?

\frac{3\pi\sqrt{2}}{2}

2\sqrt{2} + \frac{3\pi\sqrt{2}}{2}

4\sqrt{2} + \frac{3\pi\sqrt{2}}{2}

4 + \frac{3\pi}{2}

8 + 3\pi

Show answer & explanation

Answer: $4\sqrt{2} + \frac{3\pi\sqrt{2}}{2}$

Answer

The perimeter of the sector is

4\sqrt{2} + \frac{3\pi\sqrt{2}}{2}

inches.

The area of a sector is given by

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

. Substituting the given area

3\pi

and the central angle

\frac{3\pi}{4}

, we get

3\pi = \frac{3\pi}{8} r^2

, which simplifies to

r^2 = 8

and

r = 2\sqrt{2}

. The perimeter of the sector consists of the curved arc length

s = r\theta

plus the two straight radii of the sector (

2r

). Substituting the values yields a perimeter of

2(2\sqrt{2}) + (2\sqrt{2})\left(\frac{3\pi}{4}\right) = 4\sqrt{2} + \frac{3\pi\sqrt{2}}{2}

.

Step-by-Step Solution

1

Use the sector area formula in radians,

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

, to solve for the radius

r

of the circle.

3\pi = \frac{1}{2} r^2 \left(\frac{3\pi}{4}\right) \implies 3\pi = \frac{3\pi}{8} r^2 \implies r^2 = 8 \implies r = 2\sqrt{2}

inches.

Finding the radius of the circle is necessary to calculate both the arc length and the straight boundary segments of the sector.

2

Calculate the arc length

s

of the sector using the formula

s = r\theta

.

s = (2\sqrt{2})\left(\frac{3\pi}{4}\right) = \frac{3\pi\sqrt{2}}{2}

inches.

The arc length represents the curved boundary of the sector.

3

Compute the total perimeter of the sector by adding the arc length to the two boundary radii:

P = 2r + s

.

P = 2(2\sqrt{2}) + \frac{3\pi\sqrt{2}}{2} = 4\sqrt{2} + \frac{3\pi\sqrt{2}}{2}

inches.

The perimeter of a sector is the sum of the curved arc length and the two straight radial boundary segments.

Key Concept

Calculating the perimeter of a circular sector using its area and central angle in radians.

Estimated Time:1m 35s

Question 13Question →

A sector of a circle with radius $r$ inches has a central angle measuring $60^\circ$ . The ratio of the area of the sector (in square inches) to the perimeter of the sector (in inches) is $3:1$ . What is the radius, $r$ , of the circle, in inches?

\frac{36 + 6\pi}{\pi}

6

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

\frac{18 + 6\pi}{\pi}

\frac{18 + 3\pi}{\pi}

Show answer & explanation

Answer: $\frac{36 + 6\pi}{\pi}$

Answer

The radius of the circle is

\frac{36 + 6\pi}{\pi}

inches.

The sector area is

\frac{1}{6}\pi r^2

and the total sector perimeter (arc length plus two radii) is

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

. Setting the area equal to 3 times the perimeter gives

\frac{1}{6}\pi r^2 = 6r + \pi r

. Dividing by non-zero

r

gives

\frac{1}{6}\pi r = 6 + \pi

, which yields

r = \frac{36 + 6\pi}{\pi}

.

Step-by-Step Solution

1

Express the sector area in terms of radius

r

.

Sector Area

= \frac{60^\circ}{360^\circ} \times \pi r^2 = \frac{1}{6}\pi r^2

The area of a sector with central angle

\theta

in degrees is

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

.

2

Express the sector arc length and total sector perimeter in terms of

r

.

Arc Length

= \frac{60^\circ}{360^\circ} \times 2\pi r = \frac{\pi r}{3}

; Sector Perimeter

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

The total perimeter of a sector includes the curved arc length plus the two straight radii that enclose it.

3

Set up the equation using the given ratio of Area to Perimeter (

3:1

).

\frac{\frac{1}{6}\pi r^2}{2r + \frac{\pi r}{3}} = 3 \implies \frac{1}{6}\pi r^2 = 3\left(2r + \frac{\pi r}{3}\right)

A ratio of

3:1

means Area

= 3 \times \text{Perimeter}

.

4

Solve the equation for

r

.

\frac{1}{6}\pi r^2 = 6r + \pi r \implies \frac{1}{6}\pi r = 6 + \pi \implies r = \frac{6(6 + \pi)}{\pi} = \frac{36 + 6\pi}{\pi}

Dividing both sides by

r

(since

r > 0

) simplifies the quadratic relationship to a linear equation in

r

.

Key Concept

Calculating sector area, arc length, and sector perimeter using central angle ratios.

Estimated Time:2m 0s

Question 14Question →

The tip of a mechanical pendulum swings along a circular arc, sweeping out a sector of a circle. The arc length traveled by the tip of the pendulum is $8\pi$ inches, and the area of the circular sector swept out is $48\pi$ square inches. What is the total perimeter, in inches, of this circular sector?

12 + 8\pi

24 + 8\pi

48 + 8\pi

8\pi

24 + 4\pi

Show answer & explanation

Answer: $24 + 8\pi$

Answer

The total perimeter of the circular sector is

24 + 8\pi

inches.

The area of a circular sector with radius

r

and arc length

s

is given by

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

. Substituting the given values

A = 48\pi

and

s = 8\pi

gives

48\pi = \frac{1}{2} r (8\pi)

, which simplifies to

48\pi = 4\pi r

, so

r = 12

inches. The total perimeter of the sector consists of the arc length plus two straight radii:

P = 2r + s = 2(12) + 8\pi = 24 + 8\pi

inches.

Step-by-Step Solution

1

Relate sector area, radius, and arc length

Use the formula

\text{Area} = \frac{1}{2} r s

, where

s

is the arc length and

r

is the radius.

This direct relationship allows solving for the radius without needing to calculate the central angle explicitly.

2

Solve for the radius

r

48\pi = \frac{1}{2} \cdot r \cdot (8\pi) \implies 48\pi = 4\pi r \implies r = 12

inches.

Dividing both sides by

4\pi

determines the length of the pendulum arm (the radius).

3

Calculate the total sector perimeter

\text{Perimeter} = 2r + s = 2(12) + 8\pi = 24 + 8\pi$ inches.

The boundary of a circular sector consists of two straight radii and the curved arc.

Key Concept

Relationship between Sector Area, Arc Length, Radius, and Sector Perimeter

Estimated Time:1m 30s

Question 15Question →

A decorative emblem is shaped as the region bounded by two concentric circular sectors sharing the same central angle of $\theta$ radians. The outer sector has radius $R\text{ cm}$ , and the inner sector has radius $r\text{ cm}$ , where the difference between the two radii is $R - r = 4\text{ cm}$ . If the area of the emblem is $15\pi\text{ cm}^2$ and the length of the outer arc is $5\pi\text{ cm}$ , what is the value of $\theta$ , in radians?

\frac{3\pi}{8}

\frac{5\pi}{16}

\frac{5\pi}{8}

\frac{3\pi}{4}

\frac{5\pi}{4}

Show answer & explanation

Answer: $\frac{5\pi}{8}$

Answer

The central angle

\theta

is

\frac{5\pi}{8}

radians.

Using the formulas for arc length (

s = R\theta

) and sector area (

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

), the area of the emblem is

A = \frac{1}{2}\theta(R^2 - r^2) = \frac{1}{2}\theta(R - r)(R + r)

. Substituting

R - r = 4\text{ cm}

gives

15\pi = 2\theta(R + r)

. Since

R\theta = 5\pi

, we have

R + r = \frac{10\pi}{\theta} - 4

. Substituting this into the area equation yields

2\theta\left(\frac{10\pi}{\theta} - 4\right) = 15\pi

, which simplifies to

20\pi - 8\theta = 15\pi

, yielding

\theta = \frac{5\pi}{8}

.

Step-by-Step Solution

1

Express the outer arc length using the radian arc length formula.

s_{\text{outer}} = R\theta = 5\pi

, which gives

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

.

Arc length in radians is given by

s = r\theta

.

2

Express the area of the emblem as the difference between the outer and inner sector areas.

A = \frac{1}{2}R^2\theta - \frac{1}{2}r^2\theta = \frac{1}{2}\theta(R^2 - r^2) = 15\pi

.

The emblem is formed by removing the inner sector from the outer sector.

3

Factor

R^2 - r^2

as

(R - r)(R + r)

and substitute

R - r = 4

.

15\pi = \frac{1}{2}\theta(4)(R + r) = 2\theta(R + r)

.

The difference of squares allows substituting the known difference between radii.

4

Substitute

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

and

r = \frac{5\pi}{\theta} - 4

into the sum

(R + r)

.

R + r = \frac{10\pi}{\theta} - 4

.

Expreing

R + r

solely in terms of

\theta

allows solving a single variable equation.

5

Solve the resulting equation for

\theta

.

2\theta \left(\frac{10\pi}{\theta} - 4\right) = 15\pi \implies 20\pi - 8\theta = 15\pi \implies 8\theta = 5\pi \implies \theta = \frac{5\pi}{8}

.

Distributing

2\theta

cancels

\theta

in the first term and leaves a linear equation in

\theta

.

Key Concept

Sector area and arc length formulas in radians applied to concentric regions

Estimated Time:2m 0s

Question 16Question →

A circular stained-glass window with a radius of $12\text{ inches}$ contains a sector, $OAB$ , with a central angle measuring $120^\circ$ . A straight piece of lead wire is placed along chord $\overline{AB}$ , dividing sector $OAB$ into a triangular region $\triangle OAB$ and a circular segment bounded by chord $\overline{AB}$ and arc $\widehat{AB}$ . What is the area, in square inches, of the circular segment?

48\pi - 36\sqrt{3}

144\pi - 36\sqrt{3}

24\pi - 36\sqrt{3}

48\pi - 72\sqrt{3}

48\pi - 36

Show answer & explanation

Answer: $48\pi - 36\sqrt{3}$

Answer

48\pi - 36\sqrt{3}

square inches

The area of a circular segment is found by subtracting the area of the central triangle from the area of the sector. The sector area with a

120^\circ

central angle and a radius of

12

is

\frac{120^\circ}{360^\circ} \cdot \pi (12^2) = 48\pi

. The area of the isosceles triangle formed by the two radii and the chord is

\frac{1}{2} (12)(12) \sin(120^\circ) = 36\sqrt{3}

. Thus, the segment area is

48\pi - 36\sqrt{3}

square inches.

Step-by-Step Solution

1

Calculate the area of sector

OAB

Sector Area =

\frac{120^\circ}{360^\circ} \cdot \pi \cdot (12)^2 = \frac{1}{3} \cdot 144\pi = 48\pi\text{ sq in}

The area of a sector with central angle

\theta

in degrees is given by

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

.

2

Calculate the area of triangle

\triangle OAB

Triangle Area =

\frac{1}{2} \cdot 12 \cdot 12 \cdot \sin(120^\circ) = 72 \cdot \frac{\sqrt{3}}{2} = 36\sqrt{3}\text{ sq in}

The area of a triangle with two sides

a

and

b

and included angle

\theta

is

\frac{1}{2} a b \sin(\theta)

.

3

Subtract the triangle area from the sector area to find the circular segment area

Segment Area =

48\pi - 36\sqrt{3}\text{ sq in}

A circular segment area is equal to the area of the containing sector minus the area of the central triangle.

Key Concept

Area of a Circular Segment

Estimated Time:2m 0s

Question 17Question →

An automated lawn sprinkler sweeps through a central angle of $\theta$ radians and waters a sector-shaped region of radius $r$ feet. Due to a mechanical adjustment, the central angle $\theta$ is increased by $20\%$ , while the water pressure is reduced such that the radius $r$ is decreased by $10\%$ . What is the net percentage change in the area of the region watered by the sprinkler?

Decreases by

2.8\%

Increases by

2.8\%

Increases by

8.0\%

Increases by

10.0\%

Decreases by

19.0\%

Show answer & explanation

Answer: Decreases by $2.8\%$

Answer

Decreases by

2.8\%

The area of a circular sector is proportional to the square of its radius and linearly proportional to its central angle (

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

). Decreasing the radius by

10\%

scales the radius by

0.90

, which scales

r^2

by

(0.90)^2 = 0.81

. Increasing the central angle by

20\%

scales

\theta

by

1.20

. The overall scaling factor for the sector area is

0.81 \times 1.20 = 0.972

. This corresponds to

97.2\%

of the original area, which is a net decrease of

2.8\%

.

Step-by-Step Solution

1

Express the original sector area in terms of

r

and

\theta

.

A_1 = \frac{1}{2} r^2 \theta

The area of a circular sector with radius

r

and central angle

\theta

in radians is given by

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

.

2

Express the new radius and central angle after the percentage adjustments.

r_{new} = 0.90 r

and

\theta_{new} = 1.20 \theta

A

10\%

decrease in radius leaves

90\%

of

r

, and a

20\%

increase in angle yields

120\%

of

\theta

.

3

Substitute the new variables into the sector area formula and simplify.

A_2 = \frac{1}{2} (0.90 r)^2 (1.20 \theta) = \frac{1}{2} (0.81 r^2) (1.20 \theta) = 0.972 \left(\frac{1}{2} r^2 \theta\right) = 0.972 A_1

Squaring

0.90

gives

0.81

, and multiplying

0.81 \times 1.20

yields

0.972

.

4

Calculate the net percentage change from

A_1

to

A_2

.

Percentage Change

= (0.972 - 1) \times 100\% = -2.8\%

A factor of

0.972

means the new area is

97.2\%

of the original area, which is a decrease of

2.8\%

.

Key Concept

Circular Sector Area under proportional changes of parameters

Estimated Time:2m 0s

Question 18Question →

In a circle with center $O$ , sector $AOB$ has an area of $18\pi$ square centimeters and an arc length along $\widehat{AB}$ of $3\pi$ centimeters. A straight line segment $AB$ is drawn to complete triangle $AOB$ . What is the area, in square centimeters, of the circular segment bounded by line segment $AB$ and arc $\widehat{AB}$ ?

18\pi - 36\sqrt{2}

18\pi - 72\sqrt{2}

18\pi - 36

36\pi - 18\sqrt{2}

18\pi - 72

Show answer & explanation

Answer: $18\pi - 36\sqrt{2}$

Answer

The area of the circular segment is

18\pi - 36\sqrt{2}

square centimeters.

The expression

18\pi - 36\sqrt{2}

correctly represents the area of the circular segment. By dividing sector area (

18\pi

) by arc length (

3\pi

), we obtain

\frac{1}{2}r = 6

, giving a radius

r = 12\text{ cm}

. Substituting

r = 12

into

r\theta = 3\pi

gives central angle

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

radians (

45^\circ

). The area of triangle

AOB

is

\frac{1}{2}(12)^2\sin(45^\circ) = 36\sqrt{2}

. Subtracting the triangle area from the sector area yields

18\pi - 36\sqrt{2}

.

Step-by-Step Solution

1

Relate sector area and arc length formulas to find radius

r

Sector area

A = \frac{1}{2}r^2\theta = 18\pi

and arc length

s = r\theta = 3\pi

. Dividing sector area by arc length gives

\frac{\frac{1}{2}r^2\theta}{r\theta} = \frac{18\pi}{3\pi} \implies \frac{1}{2}r = 6 \implies r = 12\text{ cm}

.

Dividing the sector area equation by the arc length equation isolates the radius

r

.

2

Find central angle

\theta

s = r\theta \implies 3\pi = 12\theta \implies \theta = \frac{3\pi}{12} = \frac{\pi}{4}\text{ radians}

(

45^\circ

).

Knowing the radius

r

allows calculating

\theta

directly from the arc length formula.

3

Calculate the area of triangle

AOB

\text{Area}(AOB) = \frac{1}{2}r^2\sin\theta = \frac{1}{2}(12)^2\sin\left(\frac{\pi}{4}\right) = 72 \cdot \frac{\sqrt{2}}{2} = 36\sqrt{2}\text{ cm}^2.

The area of a triangle with two sides of length

r

and included angle

\theta

is

\frac{1}{2}r^2\sin\theta

.

4

Subtract the triangle area from the sector area to find the segment area

\text{Segment Area} = \text{Area}(\text{sector } AOB) - \text{Area}(\triangle AOB) = 18\pi - 36\sqrt{2}\text{ cm}^2.

The region bounded by the chord and the arc is the sector minus the central triangle.

Key Concept

The area of a circular segment is found by subtracting the area of the central triangle (

\frac{1}{2}r^2\sin\theta

) from the area of the circular sector (

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

).

Question 19Question →

A region $ABCD$ is bounded by two concentric circular arcs of radii $R$ and $r$ (where $R > r$ ) and two radial line segments, all sharing a central angle of $72^\circ$ . The area of region $ABCD$ is $60\pi\text{ cm}^2$ , and the total perimeter of region $ABCD$ is $(12\pi + 20)\text{ cm}$ . What is the value of the outer radius $R$ , in centimeters?

10

15

20

25

30

Show answer & explanation

Answer: 20

Answer

The outer radius

R

is 20 centimeters.

The central angle of

72^\circ

represents

\frac{72}{360} = \frac{1}{5}

of a circle. The area of the region is

\frac{\pi}{5}(R^2 - r^2) = 60\pi

, which simplifies to

R^2 - r^2 = 300

, or

(R+r)(R-r) = 300

. The total perimeter is the sum of the outer arc

\frac{2\pi R}{5}

, the inner arc

\frac{2\pi r}{5}

, and the two straight side segments

2(R-r)

. Setting

\frac{2\pi}{5}(R+r) + 2(R-r) = 12\pi + 20

yields

\frac{2}{5}(R+r) = 12 \implies R+r = 30

and

2(R-r) = 20 \implies R-r = 10

. Solving the system

R+r = 30

and

R-r = 10

by adding the equations gives

2R = 40

, so

R = 20\text{ cm}

.

Step-by-Step Solution

1

Determine the fraction of the full circle represented by the

72^\circ

central angle.

The fraction is

\frac{72^\circ}{360^\circ} = \frac{1}{5}

.

Arc lengths and sector areas are proportional to the ratio of the central angle to

360^\circ

.

2

Set up and simplify the equation for the area of the region

ABCD

.

\frac{1}{5}\pi R^2 - \frac{1}{5}\pi r^2 = 60\pi \implies R^2 - r^2 = 300 \implies (R+r)(R-r) = 300

.

The region's area is the difference between the outer sector area and inner sector area.

3

Set up and simplify the equation for the perimeter of region

ABCD

.

\frac{1}{5}(2\pi R) + \frac{1}{5}(2\pi r) + 2(R - r) = 12\pi + 20 \implies \frac{2\pi}{5}(R + r) + 2(R - r) = 12\pi + 20

.

The perimeter consists of the outer arc, the inner arc, and two radial segments each of length

R - r

.

4

Equate corresponding rational and

\pi

-coefficient terms to solve for

(R+r)

and

(R-r)

.

\frac{2}{5}(R + r) = 12 \implies R + r = 30

, and

2(R - r) = 20 \implies R - r = 10

.

Equating the algebraic components yields a system of linear equations.

5

Solve the system of equations for the outer radius

R

.

Adding the equations gives

(R + r) + (R - r) = 30 + 10 \implies 2R = 40 \implies R = 20\text{ cm}

.

Eliminating

r

isolates the required variable

R

.

Key Concept

Arc length and sector area of concentric circular regions

Estimated Time:2m 0s

Question 20Question →

In a circle with center $O$ , a sector has a central angle measuring $\frac{2\pi}{5}$ radians and an arc length of $6\pi$ centimeters. What is the area, in square centimeters, of the sector?

15\pi

\frac{9\pi}{5}

45\pi

90\pi

225\pi

Show answer & explanation

Answer: $45\pi$

Answer

The area of the sector is

45\pi

square centimeters.

To find the area of the sector, first determine the radius

r

of the circle using the arc length formula for radians,

s = r\theta

. Substituting

s = 6\pi

and

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

yields

6\pi = r \left(\frac{2\pi}{5}\right)

, which simplifies to

r = 15

centimeters. Then, apply the sector area formula

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

. Substituting

r = 15

and

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

gives

A = \frac{1}{2} (15^2) \left(\frac{2\pi}{5}\right) = 45\pi

square centimeters. Alternatively, using

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

directly gives

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

square centimeters.

Step-by-Step Solution

1

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

r = 15

centimeters

The arc length formula for an angle in radians is

s = r\theta

. Substituting

s = 6\pi

and

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

gives

6\pi = r \left(\frac{2\pi}{5}\right)

, which yields

r = 6\pi \cdot \frac{5}{2\pi} = 15

cm.

2

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

A = 45\pi

square centimeters

The area of a sector with central angle

\theta

in radians is

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

. Substituting

r = 15

and

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

gives

A = \frac{1}{2} (15^2) \left(\frac{2\pi}{5}\right) = \frac{1}{2} (225) \left(\frac{2\pi}{5}\right) = 45\pi

square centimeters.

Key Concept

Arc Length and Sector Area in Radians

Estimated Time:1m 0s

Circle Geometry: Arc Length and Sector Area

Answer

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