Circles Practice Questions — Armed Services Vocational Aptitude Battery (ASVAB) — Page 2

Question 21Question →

A searchlight beam sweeps horizontally across a central angle of $72^\circ$ . If the maximum effective range of the searchlight is $20 \text{ meters}$ , what is the length of the arc formed by the outer edge of the beam? (Leave your answer in terms of $\pi$ ).

4\pi \text{ meters}

8\pi \text{ meters}

40\pi \text{ meters}

80\pi \text{ meters}

Show answer & explanation

Answer: $8\pi \text{ meters}$

Answer

8\pi \text{ meters}

The length of an arc is calculated by multiplying the full circumference of the circle (

2\pi r

) by the fraction of the circle represented by the central angle (

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

). In this scenario, the full circumference is

2\pi(20) = 40\pi \text{ meters}

. The fraction representing the

72^\circ

sweep is

\frac{72}{360}

, which simplifies to

\frac{1}{5}

. Multiplying the full circumference by this fraction gives

\frac{1}{5} \times 40\pi = 8\pi \text{ meters}

.

Step-by-Step Solution

1

Identify the formula for the arc length of a circle.

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

The arc length represents a specific fraction of the entire circumference based on the central angle.

2

Determine the fraction of the circle covered by the searchlight beam.

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

Simplifying the angle fraction makes the subsequent multiplication easier.

3

Calculate the full circumference of the circle using the given radius (range).

C = 2\pi(20) = 40\pi \text{ meters}

The full circumference is necessary to find the length of the fractional arc.

4

Multiply the full circumference by the angle fraction.

\frac{1}{5} \times 40\pi = 8\pi \text{ meters}

This calculation applies the proportional fraction to the total boundary of the circle, yielding the arc length.

Key Concept

Arc Length of a Circle

Estimated Time:1m 0s

Question 22Question →

Two circular dials on a control panel have a radius ratio of $2:3$ . If the circumference of the smaller dial is $12\pi \text{ centimeters}$ , what is the circumference of the larger dial?

8\pi \text{ cm}

18\pi \text{ cm}

36\pi \text{ cm}

81\pi \text{ cm}

Show answer & explanation

Answer: $18\pi \text{ cm}$

Answer

The circumference of the larger dial is

18\pi \text{ cm}

.

The ratio of the radii is 2:3. The smaller dial has a circumference of

12\pi

, meaning its radius is

6

(since

C = 2\pi r

). Setting up the proportion

\frac{6}{r} = \frac{2}{3}

gives a larger radius of

9

. The circumference of the larger dial is then

2\pi(9) = 18\pi \text{ cm}

.

Step-by-Step Solution

1

Determine the radius of the smaller dial.

2\pi r = 12\pi \rightarrow r = 6 \text{ cm}

The radius of the smaller dial is required to use the given ratio.

2

Apply the ratio to find the radius of the larger dial.

\frac{6}{r_{large}} = \frac{2}{3} \rightarrow 2r_{large} = 18 \rightarrow r_{large} = 9 \text{ cm}

The problem states the radius ratio between the smaller and larger dials is 2:3.

3

Calculate the circumference of the larger dial.

C = 2\pi(9) = 18\pi \text{ cm}

The question asks for the circumference of the larger dial.

Key Concept

Using proportions to find unknown circle dimensions and calculating circumference.

Alternative Method

Because circumference is directly proportional to radius (

C = 2\pi r

), the circumferences of two circles share the exact same ratio as their radii (

2:3

). Therefore, you can set up the proportion

\frac{12\pi}{C} = \frac{2}{3}

to solve directly for

C = 18\pi

without needing to calculate the specific radii.

Estimated Time:1m 0s

Question 23Question →

A circle with center $O$ has points $A$ , $B$ , $C$ , and $D$ on its circumference.

• Line segment $AB$ passes through the center $O$ .
• Line segment $CD$ does not pass through $O$ .
• Line $L$ intersects the circle exactly at point $A$ .
• Line $M$ intersects the circle at points $C$ and $B$ , and extends infinitely in both directions.

Match each geometric term on the left with the notation that represents it on the right.

Click a left item, then click its matching right item

Show answer & explanation

Answer

Radius corresponds to Line segment OA; Diameter corresponds to Line segment AB; Chord corresponds to Line segment CD; Tangent corresponds to Line L; Secant corresponds to Line M.

Each geometric term strictly defines a specific type of line or segment related to a circle. By systematically checking the endpoints and intersection points described in the stem, we accurately pair each term to its corresponding notation.

Step-by-Step Solution

1

Evaluate Line segment OA

Matches Radius

Since O is the center and A is on the circle, the segment connecting them is the radius.

2

Evaluate Line segment AB

Matches Diameter

The segment connects two points on the circle and passes through the center O.

3

Evaluate Line segment CD

Matches Chord

The segment connects two points on the circle but does not pass through the center.

4

Evaluate Line L

Matches Tangent

The line touches the circle at exactly one point, which is the definition of a tangent line.

5

Evaluate Line M

Matches Secant

The line extends infinitely and intersects the circle at exactly two points.

Key Concept

Geometric definitions of circle components

Question 24Question →

A circular steel plate is cut into $6$ equal sectors. If the total area of the original plate was $144\pi \text{ cm}^2$ , what is the arc length of the outer edge of one sector?

4\pi \text{ cm}

2\pi \text{ cm}

24\pi \text{ cm}

12\pi \text{ cm}

Show answer & explanation

Answer: $4\pi \text{ cm}$

Answer

The arc length of one sector is

4\pi \text{ cm}

.

The correct answer is found by first using the total area (

144\pi

) to find the radius (

r = 12

). Next, the full circumference is calculated using

C = 2\pi r

, resulting in

24\pi

. Finally, because the plate is divided into 6 equal sectors, the full circumference is divided by 6, giving an arc length of

4\pi \text{ cm}

.

Step-by-Step Solution

1

Find the radius of the original circular plate.

Using the area formula

A = \pi r^2

, substitute the given area:

144\pi = \pi r^2

. Dividing by

\pi

gives

r^2 = 144

, so the radius

r = 12 \text{ cm}

.

The radius is needed to calculate the total circumference of the circle.

2

Calculate the full circumference of the plate.

Using the circumference formula

C = 2\pi r

, substitute

r = 12

to get

C = 2\pi(12) = 24\pi \text{ cm}

.

The arc length of a sector is a fraction of the circle's total circumference.

3

Determine the arc length of one of the 6 equal sectors.

Divide the full circumference by 6:

\frac{24\pi}{6} = 4\pi \text{ cm}

.

Since the plate is cut into 6 equal sectors, the outer edge of one sector is exactly one-sixth of the total circumference.

Key Concept

The relationship between a circle's area, its circumference, and the proportional arc length of its sectors.

Question 25Question →

The circular foundation of a radar dome has a circumference of $36\pi \text{ feet}$ . What is the area of this circular foundation, in square feet?

(Enter only the numerical coefficient of $\pi$ in your answer. For example, if the area is $100\pi$ , enter 100.)

Show answer & explanation

Answer: 324

Answer

324

The circumference of a circle is calculated using

C = 2\pi r

. Setting the given circumference

36\pi = 2\pi r

gives a radius of

18 \text{ feet}

. The area is found using

A = \pi r^2

, which becomes

A = \pi(18)^2 = 324\pi \text{ square feet}

. Since the question asks for the numerical coefficient of

\pi

, the correct value to enter is 324.

Step-by-Step Solution

1

Find the radius of the circular foundation from its given circumference.

r = 18

The circumference formula is

C = 2\pi r

. Setting

2\pi r = 36\pi

allows us to solve for the radius.

2

Calculate the area of the foundation using the radius.

A = 324\pi

The area formula is

A = \pi r^2

. Substituting

r = 18

gives

A = \pi(18)^2 = 324\pi

.

3

Extract the numerical coefficient of

\pi

for the final answer.

324

The question explicitly asks to enter only the numerical coefficient of

\pi

.

Key Concept

Calculating the area of a circle by first deriving the radius from its circumference

Circles

Answer

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Key Concept

Common Mistakes

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Common Mistakes

Alternative Method

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Common Mistakes

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Common Mistakes

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Common Mistakes