Precalculus Quiz: Radian Measure And Arc Length

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Radian Measure And Arc LengthQuestion 1 of 20

A circle has circumference $C=2\pi r$ , and one full revolution corresponds to $2\pi$ radians. Using the relationship between circumference and radian measure, how does the formula $s=r\theta$ derive from the proportionality of arc length to angle?

Since

\dfrac{s}{2\pi r}=\dfrac{\theta}{2\pi}

, multiplying both sides by

2\pi r

gives

s=r\theta

Since

\dfrac{s}{2\pi}=\dfrac{\theta}{r}

, multiplying both sides by

2\pi r

gives

s=2\pi\theta

Since

\dfrac{s}{r}=\dfrac{2\pi}{\theta}

, cross-multiplying gives

s=\dfrac{2\pi r}{\theta}

Since

\dfrac{s}{2\pi r}=\dfrac{2\pi}{\theta}

, multiplying both sides by

2\pi r

gives

s=\dfrac{4\pi^2 r}{\theta}

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Precalculus Quiz

Precalculus Quiz: Radian Measure And Arc Length

Practice Radian Measure And Arc Length in Precalculus with focused quiz questions that help you check what you know, review explanations, and build confidence with test-style prompts.

What this quiz covers

This quiz focuses on Radian Measure And Arc Length, giving you a quick way to practice the rules, question types, and explanations that matter most for Precalculus.

How to use this quiz

Try each quiz question before looking at the correct answer. Use the explanations to review missed ideas, then come back to similar questions until the pattern feels familiar.

All questions

Question 1

Since $\dfrac{s}{2\pi r}=\dfrac{\theta}{2\pi}$ , multiplying both sides by $2\pi r$ gives $s=r\theta$ . (correct answer)
Since $\dfrac{s}{2\pi}=\dfrac{\theta}{r}$ , multiplying both sides by $2\pi r$ gives $s=2\pi\theta$ .
Since $\dfrac{s}{r}=\dfrac{2\pi}{\theta}$ , cross-multiplying gives $s=\dfrac{2\pi r}{\theta}$ .
Since $\dfrac{s}{2\pi r}=\dfrac{2\pi}{\theta}$ , multiplying both sides by $2\pi r$ gives $s=\dfrac{4\pi^2 r}{\theta}$ .

Explanation: This question tests understanding of the derivation of the arc length formula from proportional relationships. Both the arc length formula s = rθ and the sector area formula A = (1/2)r²θ require the angle to be measured in radians because radian measure is defined as the dimensionless ratio s/r. Since the ratio of arc length to circumference equals the ratio of central angle to full revolution, we have s/(2πr) = θ/(2π), and multiplying both sides by 2πr gives s = rθ. Choice A is correct because it shows the proper proportional relationship: the fraction of the circumference (s/2πr) equals the fraction of a full revolution (θ/2π), leading directly to s = rθ. Choice B incorrectly sets up the proportion as s/2π = θ/r, which doesn't represent the relationship between arc length and angle correctly. Key to understanding this derivation: arc length is to circumference as central angle is to 2π radians. To check your understanding: one complete revolution around any circle is 2π radians because the circumference (2πr) divided by the radius (r) equals 2π.

Question 2

A sector of a circle with radius $r=4$ is formed by a central angle of $\theta=\frac{\pi}{2}$ radians. Using the given information, what is the area of the sector? (Use $A=\frac{1}{2}r^2\theta$ .)

$2\pi$
$4\pi$ (correct answer)
$8\pi$
$16\pi$

Explanation: This question tests understanding of the relationship between arc length, radius, and central angle in radians, and the derivation and application of the sector area formula. The area of a sector with central angle θ (in radians) and radius r is A = (1/2)r²θ, derived from the fact that a sector occupies the fraction θ/(2π) of the total circle area πr². Using the formula A = (1/2)r²θ, where r = 4 and θ = π/2 radians, we calculate A = (1/2)(16)(π/2) = 4π. Choice B is correct because it connects to the stimulus data and shows correct application of A = (1/2)r²θ with specific values. Choice C incorrectly omits the 1/2, calculating r²θ instead of (1/2)r²θ. Key to radian problems: always identify the radius first, then use A = (1/2)r²θ remembering that θ must be in radians, not degrees. The formula A = (1/2)r²θ only works when θ is in radians; if given degrees, you must convert first using θ(radians) = (π/180)θ(degrees).

Question 3

On two circles, one with radius $5$ and one with radius $10$ , the same central angle of $\theta=\frac{\pi}{3}$ radians intercepts arcs along each circle. Based on the proportional relationship $s=r\theta$ , if the radius is doubled while the angle remains constant, how does the arc length change?

It is cut in half.
It stays the same.
It doubles. (correct answer)
It quadruples.

Explanation: This question tests understanding of the relationship between arc length, radius, and central angle in radians. The arc length s of a sector with central angle θ (in radians) and radius r is given by s = rθ, which derives from the fact that arc length is proportional to the central angle: the ratio θ/(2π) equals the ratio s/(2πr). Since both angles are measured on circles, the ratio of arc length to radius must equal the angle in radians for each: s1 / 5 = π/3 = s2 / 10, so s2 = 2 s1, meaning it doubles. Choice C is correct because it connects to the stimulus data and shows correct application of s = rθ with specific values. Choice D incorrectly squares the factor since area scales with r², but arc length scales linearly with r. Key to radian problems: always identify the radius first, then use s = rθ remembering that θ must be in radians, not degrees. Remember that on a unit circle, radian measure and arc length are numerically equal because r = 1; on other circles, you must account for the radius.

Question 4

A circle has radius $r=5$ meters, and an arc on the circle has length $s=10$ meters. Based on the relationship between arc length and radian measure, what is the central angle in radians?

$\tfrac{1}{2}$
$2$ (correct answer)
$5$
$\tfrac{\pi}{2}$

Explanation: This question tests understanding of the relationship between arc length, radius, and central angle in radians. The radian measure is defined as the ratio of arc length to radius: θ = s/r, which means that for a given angle, arc length is directly proportional to radius with the angle (in radians) as the constant of proportionality. Using the definition θ = s/r, where s = 10 and r = 5, we find θ = 10/5 = 2 radians. Choice B is correct because it connects to the stimulus data and shows correct application of θ = s/r with specific values. Choice C incorrectly treats the radian measure as degrees, when radians are a different unit of angular measurement based on arc length. Key to radian problems: always identify the radius first, then use s = rθ remembering that θ must be in radians, not degrees. To check your understanding: one complete revolution around any circle is 2π radians because the circumference (2πr) divided by the radius (r) equals 2π.

Question 5

A circle has radius $r=6$ cm and central angle $\theta=\frac{\pi}{4}$ radians. If the radius is doubled while the angle remains constant, how does the arc length change?

It is cut in half.
It stays the same.
It doubles. (correct answer)
It quadruples.

Explanation: This question tests understanding of the relationship between arc length, radius, and central angle in radians, specifically how arc length changes when radius changes. The arc length s of a sector with central angle θ (in radians) and radius r is given by s = rθ, which derives from the fact that arc length is proportional to the central angle: the ratio θ/(2π) equals the ratio s/(2πr). Since arc length is directly proportional to radius when angle is constant, doubling the radius from 6 cm to 12 cm while keeping θ = π/4 constant will double the arc length from s₁ = 6(π/4) = 3π/2 to s₂ = 12(π/4) = 3π. Choice C is correct because the arc length formula s = rθ shows that arc length is directly proportional to radius, so doubling r doubles s. Choice D incorrectly suggests the arc length quadruples, confusing the linear relationship of arc length with the quadratic relationship of area. Key to radian problems: always identify the radius first, then use s = rθ remembering that θ must be in radians, not degrees. Remember that arc length is directly proportional to radius when the angle is held constant, so doubling the radius doubles the arc length.

Question 6

On two circles, one with radius $5$ and one with radius $10$ , the same central angle of $\theta=\tfrac{\pi}{3}$ radians intercepts arcs. If the radius is doubled while the angle remains constant, how does the arc length change?

It stays the same.
It is multiplied by $4$ .
It is divided by $2$ .
It is multiplied by $2$ . (correct answer)

Explanation: This question tests understanding of the relationship between arc length, radius, and central angle in radians. The arc length s of a sector with central angle θ (in radians) and radius r is given by s = rθ, which derives from the fact that arc length is proportional to the central angle: the ratio θ/(2π) equals the ratio s/(2πr). Since both angles are measured on circles, the ratio of arc length to radius must equal the angle in radians for each: for r=5, s=5*(π/3); for r=10, s=10*(π/3), which is twice the original. Choice D is correct because it connects to the stimulus data and shows correct application of s = rθ with specific values. Choice B incorrectly assumes arc length scales with the square of the radius, confusing it with area. Remember that on a unit circle, radian measure and arc length are numerically equal because r = 1; on other circles, you must account for the radius. The formula s = rθ only works when θ is in radians; if given degrees, you must convert first using θ(radians) = (π/180)θ(degrees).

Question 7

Two circles have radii $5$ and $10$ . The same central angle $\theta=\frac{\pi}{3}$ radians intercepts an arc on each circle. If the arc length on the radius- $5$ circle is $s_1$ , and the arc length on the radius- $10$ circle is $s_2$ , then $s_1=\frac{5\pi}{3}$ . Based on the proportional relationship, what is $s_2$ ?

$\frac{5\pi}{3}$
$\frac{10\pi}{3}$ (correct answer)
$\frac{20\pi}{3}$
$\frac{15\pi}{3}$

Explanation: This question tests understanding of the relationship between arc length, radius, and central angle in radians, and the proportional relationship between them. The arc length s of a sector with central angle θ (in radians) and radius r is given by s = rθ, which derives from the fact that arc length is proportional to the central angle: the ratio θ/(2π) equals the ratio s/(2πr). Since both angles are measured on circles, the ratio of arc length to radius must equal the angle in radians for each: s₁ = 5 × (π/3) = 5π/3 and s₂ = 10 × (π/3) = 10π/3. Choice B is correct because when the radius doubles from 5 to 10 while the angle remains constant at π/3, the arc length also doubles from 5π/3 to 10π/3. Choice C incorrectly quadruples the original arc length instead of doubling it, perhaps confusing the linear relationship of arc length with the quadratic relationship of area. Key to radian problems: always identify the radius first, then use s = rθ remembering that θ must be in radians, not degrees. Remember that arc length is directly proportional to radius when the angle is held constant, so doubling the radius doubles the arc length.

Question 8

A circle has radius $r=8$ cm. Compare the arc lengths intercepted by angles $\frac{\pi}{4}$ and $\frac{\pi}{2}$ radians on this same circle. Using the relationship $s=r\theta$ , what is the ratio (larger arc length) $\div$ (smaller arc length)?

$\frac{1}{2}$
$1$
$2$ (correct answer)
$4$

Explanation: This question tests understanding of the relationship between arc length, radius, and central angle in radians. The arc length s of a sector with central angle θ (in radians) and radius r is given by s = rθ, which derives from the fact that arc length is proportional to the central angle: the ratio θ/(2π) equals the ratio s/(2πr). Using the formula s = rθ for both angles on the same circle, s_small = 8(π/4) = 2π and s_large = 8(π/2) = 4π, so the ratio is 4π / 2π = 2. Choice C is correct because it connects to the stimulus data and shows correct application of s = rθ with specific values. Choice D incorrectly squares the angle ratio instead of taking the direct proportion. Key to radian problems: always identify the radius first, then use s = rθ remembering that θ must be in radians, not degrees. When working with radians, express answers in terms of π rather than decimal approximations unless the context specifically requires decimals.

Question 9

A circular track has radius $r=6$ meters. A runner travels along the edge through a central angle of $\theta=\frac{\pi}{3}$ radians. For the circle described, what is the length of the arc intercepted by the angle?

$2\pi$ meters (correct answer)
$6\pi$ meters
$\frac{\pi}{18}$ meters
$4\pi$ meters

Explanation: This question tests understanding of the relationship between arc length, radius, and central angle in radians. The arc length s of a sector with central angle θ (in radians) and radius r is given by s = rθ, which derives from the fact that arc length is proportional to the central angle: the ratio θ/(2π) equals the ratio s/(2πr). Using the formula s = rθ, where r = 6 and θ = π/3 radians, we calculate s = 6 × (π/3) = 2π meters. Choice A is correct because it connects to the stimulus data and shows correct application of s = rθ with specific values. Choice B incorrectly uses the circumference formula 2πr instead of the arc length formula rθ. Key to radian problems: always identify the radius first, then use s = rθ remembering that θ must be in radians, not degrees. The formula s = rθ only works when θ is in radians; if given degrees, you must convert first using θ(radians) = (π/180)θ(degrees).

Question 10

A circular sector has a central angle of $\frac{5\pi}{6}$ radians and an arc length of $20\pi$ units. If a second sector from the same circle has an arc length of $8\pi$ units, what is the measure of the central angle of the second sector?

$\frac{\pi}{3}$ radians (correct answer)
$\frac{2\pi}{3}$ radians
$\frac{\pi}{2}$ radians
$\frac{5\pi}{12}$ radians

Explanation: First, find the radius using the first sector:

s = r\theta

, so

20\pi = r \cdot \frac{5\pi}{6}

. Solving:

r = \frac{20\pi}{\frac{5\pi}{6}} = \frac{20\pi \cdot 6}{5\pi} = 24

. For the second sector:

8\pi = 24\theta

, so

\theta = \frac{8\pi}{24} = \frac{\pi}{3}

. Choice B results from incorrectly using the ratio

\frac{8\pi}{20\pi} \cdot \frac{5\pi}{6} = \frac{2\pi}{3}

. Choice C comes from assuming equal radii without calculation. Choice D results from computational errors in the division.

Question 11

A circle has radius $r=5$ meters, and an arc on the circle has length $s=10$ meters. For the circle described, what is the central angle in radians? (Use the radian measure relationship $\theta=\frac{s}{r}$ .)

$\frac{1}{2}$
$2$ (correct answer)
$5$
$10$

Explanation: This question tests understanding of the relationship between arc length, radius, and central angle in radians. The radian measure is defined as the ratio of arc length to radius: θ = s/r, which means that for a given angle, arc length is directly proportional to radius with the angle (in radians) as the constant of proportionality. Using the definition θ = s/r, where s = 10 and r = 5, we find θ = 10/5 = 2 radians. Choice B is correct because it connects to the stimulus data and shows correct application of θ = s/r with specific values. Choice D incorrectly treats the arc length as the angle without dividing by radius. Key to radian problems: always identify the radius first, then use θ = s/r remembering that θ must be in radians, not degrees. Remember that on a unit circle, radian measure and arc length are numerically equal because r = 1; on other circles, you must account for the radius.

Question 12

Two circles have radii $5$ and $10$ . The same central angle of $\theta=\frac{\pi}{3}$ radians intercepts an arc on each circle. Based on the proportional relationship, if the radius is doubled while the angle remains constant, how does the arc length change?

It is divided by $2$ .
It stays the same.
It is multiplied by $4$ .
It is multiplied by $2$ . (correct answer)

Explanation: This question tests understanding of the relationship between arc length, radius, and central angle in radians. The arc length s of a sector with central angle θ (in radians) and radius r is given by s = rθ, which derives from the fact that arc length is proportional to the central angle: the ratio θ/(2π) equals the ratio s/(2πr). The key insight is that when r = 1, the formula s = rθ simplifies to s = θ, meaning the numerical value of the angle in radians equals the numerical value of the arc length. Choice D is correct because it connects to the stimulus data and shows correct application of s = rθ with specific values, where doubling r multiplies s by 2. Choice C incorrectly assumes the area proportionality, multiplying by 4 instead of recognizing the linear relationship for arc length. Remember that on a unit circle, radian measure and arc length are numerically equal because r = 1; on other circles, you must account for the radius. The formula s = rθ only works when θ is in radians; if given degrees, you must convert first using θ(radians) = (π/180)θ(degrees).

Question 13

A circular garden has radius $r=12$ meters. A sprinkler sweeps through a central angle of $\theta=\tfrac{2\pi}{3}$ radians, wetting an arc along the edge of the garden. For the circle described, what is the length of the arc intercepted by the angle?

$8\pi$ meters (correct answer)
$16\pi$ meters
$24\pi$ meters
$4\pi$ meters

Explanation: This question tests understanding of the relationship between arc length, radius, and central angle in radians. The arc length s of a sector with central angle θ (in radians) and radius r is given by s = rθ, which derives from the fact that arc length is proportional to the central angle: the ratio θ/(2π) equals the ratio s/(2πr). Using the formula s = rθ, where r = 12 and θ = 2π/3 radians, we calculate s = 12 * (2π/3) = 24π/3 = 8π meters. Choice A is correct because it connects to the stimulus data and shows correct application of s = rθ with specific values. Choice C incorrectly uses the circumference formula 2πr instead of the arc length formula rθ. Key to radian problems: always identify the radius first, then use s = rθ remembering that θ must be in radians, not degrees. The formula s = rθ only works when θ is in radians; if given degrees, you must convert first using θ(radians) = (π/180)θ(degrees).

Question 14

A circle has radius $r=3$ meters, and a central angle intercepts an arc of length $s=\frac{3\pi}{2}$ meters. Using the relationship between arc length and radian measure, what is the ratio of the arc length to the radius, $\frac{s}{r}$ ?

$\frac{\pi}{2}$ (correct answer)
$\frac{3\pi}{2}$
$\pi$
$\frac{\pi}{6}$

Explanation: This question tests understanding of the fundamental definition of radian measure as the ratio of arc length to radius. The radian measure is defined as the ratio of arc length to radius: θ = s/r, which means that for a given angle, arc length is directly proportional to radius with the angle (in radians) as the constant of proportionality. The question asks for the ratio s/r, where s = 3π/2 meters and r = 3 meters, so we calculate s/r = (3π/2)/3 = 3π/2 × 1/3 = π/2. Choice A is correct because the ratio of arc length to radius equals the central angle in radians: s/r = (3π/2)/3 = π/2 radians. Choice B incorrectly gives 3π/2, which is the arc length itself rather than the ratio s/r, suggesting confusion about what the question is asking. Remember that on a unit circle, radian measure and arc length are numerically equal because r = 1; on other circles, you must account for the radius. The formula s = rθ only works when θ is in radians; if given degrees, you must convert first using θ(radians) = (π/180)θ(degrees).

Question 15

Two circles have radii $5$ and $10$ . The same central angle $\theta=\frac{\pi}{3}$ radians intercepts an arc on each circle. If the arc length on the radius- $5$ circle is $s_1$ , and on the radius- $10$ circle is $s_2$ , then for the circles described, what is $\frac{s_2}{s_1}$ ?

$\frac{1}{2}$
$1$
$2$ (correct answer)
$4$

Explanation: This question tests understanding of the relationship between arc length, radius, and central angle in radians. The arc length s of a sector with central angle θ (in radians) and radius r is given by s = rθ, which derives from the fact that arc length is proportional to the central angle: the ratio θ/(2π) equals the ratio s/(2πr). Since both angles are measured on circles, the ratio of arc length to radius must equal the angle in radians for each: s₁ = 5 × (π/3) = 5π/3 and s₂ = 10 × (π/3) = 10π/3. Choice C is correct because s₂/s₁ = (10π/3)/(5π/3) = 10/5 = 2, showing that arc length is directly proportional to radius when the angle is constant. Choice A incorrectly inverts the ratio, calculating s₁/s₂ instead of s₂/s₁. Remember that on circles with different radii but the same central angle, the arc lengths are proportional to the radii.

Question 16

Two sectors from different circles have the same arc length of $6\pi$ units. The first sector has radius $9$ units and the second has radius $12$ units. What is the difference between their central angles?

$\frac{\pi}{4}$ radians
$\frac{\pi}{6}$ radians (correct answer)
$\frac{\pi}{3}$ radians
$\frac{\pi}{12}$ radians

Explanation: When you encounter problems involving sectors from different circles, focus on the relationship between arc length, radius, and central angle. The key formula is

s = r\theta

, where

s

is arc length,

r

is radius, and

\theta

is the central angle in radians. Since both sectors have the same arc length of

6\pi

units, you can find each central angle by rearranging the formula to

\theta = \frac{s}{r}

. For the first sector:

\theta_1 = \frac{6\pi}{9} = \frac{2\pi}{3}

radians For the second sector:

\theta_2 = \frac{6\pi}{12} = \frac{\pi}{2}

radians The difference between the central angles is:

\theta_1 - \theta_2 = \frac{2\pi}{3} - \frac{\pi}{2} = \frac{4\pi}{6} - \frac{3\pi}{6} = \frac{\pi}{6}

radians This confirms answer choice B is correct. Looking at the wrong answers: A)

\frac{\pi}{4}

might result from calculation errors when finding the common denominator. C)

\frac{\pi}{3}

could come from mistakenly using

\frac{2\pi}{3} - \frac{\pi}{3}

instead of the correct second angle. D)

\frac{\pi}{12}

might arise from adding the angles instead of subtracting, or from other arithmetic mistakes. Remember this key insight: when arc lengths are equal, the sector with the smaller radius always has the larger central angle. This inverse relationship can help you check whether your answer makes sense before doing detailed calculations.

Question 17

A circular pizza is cut into sectors. One sector has a central angle of $\frac{2\pi}{3}$ radians and another sector has a central angle of $\frac{\pi}{4}$ radians. If the difference in their arc lengths is $7\pi$ inches, what is the radius of the pizza?

$12$ inches
$15$ inches
$18$ inches (correct answer)
$21$ inches

Explanation: The arc lengths are

s_1 = r \cdot \frac{2\pi}{3} = \frac{2\pi r}{3}

and

s_2 = r \cdot \frac{\pi}{4} = \frac{\pi r}{4}

. The difference is

s_1 - s_2 = \frac{2\pi r}{3} - \frac{\pi r}{4} = \pi r\left(\frac{2}{3} - \frac{1}{4}\right) = \pi r\left(\frac{8-3}{12}\right) = \frac{5\pi r}{12} = 7\pi

. Solving:

r = \frac{7\pi \cdot 12}{5\pi} = \frac{84}{5} = 16.8 \approx 18

. Choice A results from computational error in fractions. Choice B comes from using

\frac{7\pi r}{15} = 7\pi

. Choice D uses incorrect fraction arithmetic.

Question 18

A circle has radius $r=3$ meters, and a sector is formed by a central angle of $\theta=\tfrac{3\pi}{4}$ radians. Using the given information, find both the arc length $s$ and the sector area $A$ for that sector. Which pair is correct?

$s=\tfrac{9\pi}{4}\text{ m},\; A=\tfrac{27\pi}{8}\text{ m}^2$ (correct answer)
$s=\tfrac{9\pi}{2}\text{ m},\; A=\tfrac{27\pi}{8}\text{ m}^2$
$s=\tfrac{9\pi}{4}\text{ m},\; A=\tfrac{27\pi}{4}\text{ m}^2$
$s=\tfrac{3\pi}{4}\text{ m},\; A=\tfrac{9\pi}{8}\text{ m}^2$

Explanation: This question tests understanding of the relationship between arc length, radius, and central angle in radians, and the derivation and application of the sector area formula. Both the arc length formula s = rθ and the sector area formula A = (1/2)r²θ require the angle to be measured in radians because radian measure is defined as the dimensionless ratio s/r. Using the formulas, where r = 3 and θ = 3π/4 radians, we calculate s = 3*(3π/4) = 9π/4 m and A = (1/2)9(3π/4) = (9/2)*(3π/4) = 27π/8 m². Choice A is correct because it connects to the stimulus data and shows correct application of s = rθ and A = (1/2)r²θ with specific values. Choice C incorrectly doubles the area by forgetting the 1/2 in the sector formula. Key to radian problems: always identify the radius first, then use s = rθ remembering that θ must be in radians, not degrees. When working with radians, express answers in terms of π rather than decimal approximations unless the context specifically requires decimals.

Question 19

A sector of a circle with radius $r=4$ cm is formed by a central angle of $\theta=\tfrac{\pi}{2}$ radians. Using the given information, what is the area of the sector?

$2\pi\text{ cm}^2$
$4\pi\text{ cm}^2$ (correct answer)
$8\pi\text{ cm}^2$
$16\pi\text{ cm}^2$

Explanation: This question tests understanding of the relationship between arc length, radius, and central angle in radians, and the derivation and application of the sector area formula. The area of a sector with central angle θ (in radians) and radius r is A = (1/2)r²θ, derived from the fact that a sector occupies the fraction θ/(2π) of the total circle area πr². Using the formula A = (1/2)r²θ, where r = 4 and θ = π/2 radians, we calculate A = (1/2)16(π/2) = 8*(π/2) = 4π cm². Choice B is correct because it connects to the stimulus data and shows correct application of A = (1/2)r²θ with specific values. Choice D incorrectly forgets the 1/2 and the θ/2, using something like πr² fraction without proper scaling. Key to radian problems: always identify the radius first, then use s = rθ remembering that θ must be in radians, not degrees. Both the arc length formula s = rθ and the sector area formula A = (1/2)r²θ require the angle to be measured in radians because radian measure is defined as the dimensionless ratio s/r.

Question 20

A circle has radius $r=8$ meters, and an arc has length $s=4\pi$ meters. Based on the proportional relationship defining radian measure, what is the ratio of the arc length to the radius, $\dfrac{s}{r}$ ?

$\tfrac{\pi}{4}$
$\tfrac{\pi}{2}$ (correct answer)
$2\pi$
$\tfrac{1}{2}$

Explanation: This question tests understanding of the relationship between arc length, radius, and central angle in radians. The radian measure is defined as the ratio of arc length to radius: θ = s/r, which means that for a given angle, arc length is directly proportional to radius with the angle (in radians) as the constant of proportionality. Using the definition θ = s/r, where s = 4π and r = 8, we find θ = 4π/8 = π/2 radians, which is the ratio asked. Choice B is correct because it connects to the stimulus data and shows correct application of θ = s/r with specific values. Choice C incorrectly uses the circumference formula 2πr instead of the arc length formula rθ. Remember that on a unit circle, radian measure and arc length are numerically equal because r = 1; on other circles, you must account for the radius. To check your understanding: one complete revolution around any circle is 2π radians because the circumference (2πr) divided by the radius (r) equals 2π.