Plane Geometry Practice Questions — ACT — Page 11

Question 201Question →

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 202Question →

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 203Question →

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 204Question →

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 205Question →

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 206Question →

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 207Question →

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 208Question →

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 209Question →

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 210Question →

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 211Question →

The measures of the three interior angles of a triangle are in the ratio $2:3:4$ . What is the measure, in degrees, of the smallest angle of the triangle?

20

40

60

72

80

Show answer & explanation

Answer: 40

Answer

40

The interior angles of any triangle sum to

180^\circ

. Given the ratio of the angles is

2:3:4

, the total number of ratio parts is

2 + 3 + 4 = 9

. Dividing the total degree sum by the number of parts gives

\frac{180^\circ}{9} = 20^\circ

per part. Since the smallest angle corresponds to the smallest part of the ratio, we multiply

2

by

20^\circ

to get

40^\circ

.

Step-by-Step Solution

1

Find the total number of parts in the ratio.

The total number of parts is

2 + 3 + 4 = 9

.

This determines how the

180^\circ

total sum of a triangle's interior angles is partitioned.

2

Calculate the value of a single part of the ratio.

One part is equal to

\frac{180^\circ}{9} = 20^\circ

.

The sum of the interior angles of any triangle is always

180^\circ

.

3

Find the measure of the smallest angle by multiplying by the smallest part of the ratio.

The smallest angle measures

2 \times 20^\circ = 40^\circ

.

The smallest angle corresponds to the smallest number in the ratio, which is 2.

Key Concept

Ratio-based angle partitioning in triangles

Question 212Question →

In $\triangle ABC$ , the measure of angle $A$ is $40^\circ$ . Point $D$ lies on side $AC$ such that segment $BD$ bisects angle $ABC$ . If the measure of angle $BDC$ is $75^\circ$ , what is the measure, in degrees, of angle $C$ ?

Show answer & explanation

Answer: 70

Answer

The measure of angle C is 70 degrees.

The correct measure of angle

C

is found by first identifying that angle

ADB

is supplementary to angle

BDC

, giving a measure of

105^\circ

. Using the triangle angle sum theorem on triangle

ABD

, we find that angle

ABD

is

35^\circ

. Since

BD

bisects angle

ABC

, angle

DBC

is also

35^\circ

. Finally, applying the triangle angle sum theorem to triangle

BCD

, we subtract the measures of angles

DBC

(

35^\circ

) and

BDC

(

75^\circ

) from

180^\circ

to get

70^\circ

.

Step-by-Step Solution

1

Find the measure of angle

ADB

using the supplementary angle relationship with angle

BDC

.

105^\circ

Angles

ADB

and

BDC

form a linear pair along the line segment

AC

, so their sum is

180^\circ

.

2

Find the measure of angle

ABD

using the sum of interior angles in

\triangle ABD

.

35^\circ

The sum of interior angles in any triangle is

180^\circ

. Therefore, the measure of angle

ABD

is

180^\circ - (40^\circ + 105^\circ) = 35^\circ

.

3

Find the measure of angle

DBC

using the definition of an angle bisector.

35^\circ

Since segment

BD

bisects angle

ABC

, the measures of angles

ABD

and

DBC

must be equal.

4

Find the measure of angle

C

using the sum of interior angles in

\triangle BCD

.

70^\circ

The sum of interior angles in

\triangle BCD

is

180^\circ

. Therefore, the measure of angle

C

is

180^\circ - (35^\circ + 75^\circ) = 70^\circ

.

Key Concept

Using the triangle angle sum theorem and angle bisector properties to determine unknown angle measures in a geometric figure.

Question 213Question →

In right triangle $ABC$ , the hypotenuse $AC$ has a length of 13 centimeters, and leg $AB$ has a length of 5 centimeters. What is the length, in centimeters, of leg $BC$ ?

Show answer & explanation

Answer: 12

Answer

The length of leg

BC

is 12 centimeters.

Applying the Pythagorean theorem, we have

5^2 + BC^2 = 13^2

, which simplifies to

25 + BC^2 = 169

. Subtracting 25 from both sides gives

BC^2 = 144

, and taking the square root of both sides gives

BC = 12

centimeters.

Step-by-Step Solution

1

Set up the Pythagorean Theorem equation for right triangle

ABC

.

AB^2 + BC^2 = AC^2

The Pythagorean Theorem states that in a right triangle, the sum of the squares of the legs is equal to the square of the hypotenuse.

2

Substitute the given values for

AB

and

AC

into the formula.

5^2 + BC^2 = 13^2

The length of leg

AB

is given as 5 centimeters, and the length of the hypotenuse

AC

is given as 13 centimeters.

3

Solve for the unknown leg length

BC

.

BC = 12

Squaring the values gives

25 + BC^2 = 169

. Subtracting 25 from both sides yields

BC^2 = 144

. Taking the square root of both sides gives

BC = 12

.

Key Concept

Pythagorean Theorem

Alternative Method

Recognize the triangle as a standard 5-12-13 Pythagorean triple, which immediately gives the missing leg length of 12 without needing calculations.

Estimated Time:30s

Question 214Question →

In a convex polygon, the measures of the interior angles form an arithmetic progression. The smallest interior angle measures $120^\circ$ , and the common difference between consecutive interior angles is $5^\circ$ . What is the number of sides of this polygon?

Show answer & explanation

Answer: 9

Answer

The number of sides of the polygon is 9.

The sum of the interior angles of a convex polygon with

n

sides is

(n-2) \times 180^\circ

. Since the angles form an arithmetic progression with the first term

a = 120^\circ

and common difference

d = 5^\circ

, their sum is also given by the arithmetic series formula:

S_n = \frac{n}{2}[2(120) + (n-1)5] = \frac{n}{2}(5n + 235)

. Setting the two sums equal yields

\frac{n}{2}(5n + 235) = 180(n - 2)

, which simplifies to the quadratic equation

n^2 - 25n + 144 = 0

. Solving this gives

n = 9

or

n = 16

. Because the polygon is convex, every interior angle must be less than

180^\circ

. If

n = 16

, the largest angle would be

120^\circ + 15(5^\circ) = 195^\circ

, which is impossible for a convex polygon. If

n = 9

, the largest angle is

120^\circ + 8(5^\circ) = 160^\circ

, which is valid. Therefore, the number of sides must be 9.

Step-by-Step Solution

1

Set up the equation equating the geometric sum of interior angles to the arithmetic series sum.

The sum of the interior angles of a convex

n

-gon is

(n - 2) \times 180^\circ

. The sum of the arithmetic sequence of angles is

S_n = \frac{n}{2}[2(120^\circ) + (n - 1)5^\circ]

. Setting them equal gives:

\frac{n}{2}(5n + 235) = 180(n - 2)

.

This establishes the algebraic relationship between the polygon's geometric properties and the given sequence of angle measures.

2

Simplify the equation and solve the resulting quadratic equation for

n

.

5n^2 + 235n = 360n - 720 \Rightarrow 5n^2 - 125n + 720 = 0 \Rightarrow n^2 - 25n + 144 = 0

. Factoring the quadratic yields

(n - 9)(n - 16) = 0

, so

n = 9

or

n = 16

.

Solving the quadratic equation yields all mathematically possible values for the number of sides.

3

Apply the convexity constraint to determine the valid number of sides.

For a polygon to be convex, every interior angle must be less than

180^\circ

. For

n = 16

, the largest angle is

120^\circ + 15(5^\circ) = 195^\circ

, which is impossible. For

n = 9

, the largest angle is

120^\circ + 8(5^\circ) = 160^\circ

, which is valid.

The definition of a convex polygon requires all interior angles to be strictly less than

180^\circ

, which eliminates the extraneous solution of 16.

Key Concept

Sum of interior angles of a convex polygon and arithmetic progressions

Question 215Question →

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 216Question →

A surveyor is mapping a triangular park. Starting at point $A$ , she walks due east for $80$ meters to point $B$ . She then turns $120^\circ$ to her left and walks in a straight line to point $C$ , which is located directly north of point $A$ . What is the straight-line distance, in meters, from point $B$ to point $C$ ?

40

80\sqrt{2}

80\sqrt{3}

160

200

Show answer & explanation

Answer: $160$

Answer

The straight-line distance from point

B

to point

C

is

160

meters.

The surveyor's movement forms a

30^\circ\text{-}60^\circ\text{-}90^\circ

right triangle where the side opposite the

30^\circ

angle is

AB = 80

meters. The hypotenuse

BC

represents the distance from

B

to

C

. In a

30^\circ\text{-}60^\circ\text{-}90^\circ

triangle, the hypotenuse is exactly twice the length of the side opposite the

30^\circ

angle. Thus, the distance is

2 \times 80 = 160

meters.

Step-by-Step Solution

1

Determine the orientation and angles of the path.

A right triangle

ABC

with a right angle at vertex

A

.

Since the path from

A

to

B

goes due east, and

C

is directly north of

A

, the angle

\angle A

is exactly

90^\circ

.

2

Calculate the interior angle at vertex

B

.

\angle B = 60^\circ

and

\angle C = 30^\circ

.

The surveyor turns

120^\circ

to the left from the extension of the eastward segment

AB

. The interior angle is the supplement:

180^\circ - 120^\circ = 60^\circ

. The sum of angles in a triangle is

180^\circ

, so the angle at

C

is

180^\circ - (90^\circ + 60^\circ) = 30^\circ

.

3

Use special right triangle ratios to find the hypotenuse.

The distance

BC = 160

meters.

In a

30^\circ\text{-}60^\circ\text{-}90^\circ

triangle, the sides are in the ratio

1 : \sqrt{3} : 2

. The side opposite the

30^\circ

angle is

AB = 80

meters. The hypotenuse

BC

is twice the length of this side:

2 \times 80 = 160

meters.

Key Concept

Ratios of a

30^\circ\text{-}60^\circ\text{-}90^\circ

special right triangle

Estimated Time:1m 0s

Question 217Question →

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

Question 218Question →

An automated agricultural sprinkler sweeps across a section of a circular field, forming a circular sector. The area of the irrigated sector is $45\pi$ square meters, and the length of the outer boundary arc of the sector is $6\pi$ meters. What is the radius, in meters, of the circular field?

7.5

9

15

22.5

30

Show answer & explanation

Answer: 15

Answer

15 meters

Using the relationship between sector area, arc length, and radius,

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

, we substitute

A = 45\pi

and

s = 6\pi

to get

45\pi = \frac{1}{2} r (6\pi) = 3\pi r

. Solving for

r

gives

r = 15

meters.

Step-by-Step Solution

1

Recall the formulas for sector area (

A

) and arc length (

s

) in terms of radius (

r

) and central angle (

\theta

in radians).

s = r\theta

and

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

These are the fundamental geometric formulas for circular sectors.

2

Express the sector area formula in terms of arc length

s

.

A = \frac{1}{2} r (r\theta) = \frac{1}{2} r s

Substituting

s = r\theta

simplifies the calculation by eliminating the central angle

\theta

.

3

Substitute the given values

A = 45\pi

and

s = 6\pi

into the simplified formula and solve for

r

.

45\pi = \frac{1}{2} r (6\pi) \implies 45\pi = 3\pi r \implies r = 15

Dividing both sides by

3\pi

yields the radius

r = 15

meters.

Key Concept

Relationship between Sector Area, Arc Length, and Radius

Plane Geometry

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