Precalculus Quiz: Understanding Radian Measure Of Angles
20 questions · exam conditions
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Understanding Radian Measure Of AnglesQuestion 1 of 20

Using the circle described: an angle θ\theta in standard position measures 3π4\frac{3\pi}{4} radians on a unit circle (radius r=1r=1), starting from the positive xx-axis. What is the length ss of the arc intercepted by the angle?

3π4\frac{3\pi}{4}
3π2\frac{3\pi}{2}
34\frac{3}{4}
135135
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Precalculus Quiz

Precalculus Quiz: Understanding Radian Measure Of Angles

Practice Understanding Radian Measure Of Angles 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 Understanding Radian Measure Of Angles, 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

Using the circle described: an angle θ\theta in standard position measures 3π4\frac{3\pi}{4} radians on a unit circle (radius r=1r=1), starting from the positive xx-axis. What is the length ss of the arc intercepted by the angle?

  1. 3π4\frac{3\pi}{4} (correct answer)
  2. 3π2\frac{3\pi}{2}
  3. 34\frac{3}{4}
  4. 135135
Explanation: This question tests understanding of radian measure as the arc length on the unit circle. On a unit circle (radius = 1), the radian measure of an angle equals the length of the arc it intercepts. Using the formula s = rθ, where r = 1 and θ = 3π/4 radians, we calculate s = 1 * (3π/4) = 3π/4. Choice A is correct because it connects to the stimulus data and shows correct application of s = rθ with θ = 3π/4 and r = 1. Choice D incorrectly treats the radian measure as degrees, when radians are a different unit of angular measurement based on 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 2

A satellite orbits Earth in a circular path. During one portion of its orbit, it travels through an arc length equal to 16\frac{1}{6} of the Earth's circumference. If the orbital radius is 1.51.5 times Earth's radius RR, what angle does this arc subtend at the center of the orbit?

  1. π9\frac{\pi}{9} radians
  2. π3\frac{\pi}{3} radians
  3. 2π9\frac{2\pi}{9} radians (correct answer)
  4. π6\frac{\pi}{6} radians
Explanation: Earth's circumference is 2πR2\pi R, so the arc length is 16⋅2πR=πR3\frac{1}{6} \cdot 2\pi R = \frac{\pi R}{3}. The orbital radius is 1.5R1.5R. Using θ=sr\theta = \frac{s}{r}: θ=πR31.5R=πR3⋅11.5R=π4.5=2π9\theta = \frac{\frac{\pi R}{3}}{1.5R} = \frac{\pi R}{3} \cdot \frac{1}{1.5R} = \frac{\pi}{4.5} = \frac{2\pi}{9} radians. Choice A uses radius 3R3R. Choice B ignores the radius multiplier. Choice D uses the fraction 16\frac{1}{6} directly as the angle coefficient.

Question 3

On the unit circle, an arc from point AA to point BB has length 7π6\frac{7\pi}{6}. If point AA corresponds to an angle of π4\frac{\pi}{4} radians in standard position, what angle in standard position corresponds to point BB?

  1. 17π12\frac{17\pi}{12} radians (correct answer)
  2. 19π12\frac{19\pi}{12} radians
  3. 5π3\frac{5\pi}{3} radians
  4. 11π6\frac{11\pi}{6} radians
Explanation: On the unit circle, arc length equals the change in angle (in radians). Starting at π4\frac{\pi}{4} and moving through an arc of length 7π6\frac{7\pi}{6}: angle at B=π4+7π6=3π12+14π12=17π12B = \frac{\pi}{4} + \frac{7\pi}{6} = \frac{3\pi}{12} + \frac{14\pi}{12} = \frac{17\pi}{12} radians. Choice B adds an extra π6\frac{\pi}{6}. Choice C converts incorrectly to a common denominator. Choice D subtracts π4\frac{\pi}{4} from 7π6\frac{7\pi}{6} instead of adding.

Question 4

In the unit circle, if an arc of length kk subtends an angle of kk radians, and another arc of length 2k2k is drawn on a circle of radius 33, what is the ratio of the angle subtended by the second arc to the angle subtended by the first arc?

  1. 13\frac{1}{3}
  2. 32\frac{3}{2}
  3. 22
  4. 23\frac{2}{3} (correct answer)
Explanation: When you encounter arc length and angle problems, remember the fundamental relationship: arc length = radius × angle (in radians). This formula is your key to solving problems involving different circles. Let's work through this step by step. For the first arc on the unit circle (radius = 1), we have an arc length of kk that subtends an angle of kk radians. We can verify this makes sense: k=1×kk = 1 \times k, which checks out. For the second arc on a circle with radius 3, we have an arc length of 2k2k. To find the angle it subtends, we use our formula: 2k=3×θ2k = 3 \times \theta, where θ\theta is the unknown angle. Solving for θ\theta: θ=2k3\theta = \frac{2k}{3} radians. The ratio of the second angle to the first angle is: 2k3k=2k3×1k=23\frac{\frac{2k}{3}}{k} = \frac{2k}{3} \times \frac{1}{k} = \frac{2}{3} Looking at the wrong answers: Choice A (13\frac{1}{3}) likely comes from confusing which angle goes in the numerator. Choice B (32\frac{3}{2}) results from incorrectly flipping the final ratio. Choice C (2) occurs if you forget to account for the different radii and simply compare arc lengths. Remember this pattern: when comparing angles subtended by arcs on circles with different radii, longer arc lengths don't necessarily mean larger angles. The radius matters crucially in the arc length formula, so always set up your equation carefully and solve for the unknown angle first.

Question 5

A circular track has a radius of 50 meters. A runner completes 34\frac{3}{4} of one full lap, then continues for an additional distance of 25π25\pi meters along the track. What is the total angle, in radians, through which the runner has moved from the starting position?

  1. 7π4\frac{7\pi}{4} radians
  2. 5π2\frac{5\pi}{2} radians
  3. 2π2\pi radians (correct answer)
  4. 9π4\frac{9\pi}{4} radians
Explanation: When you encounter circular motion problems, the key relationship to remember is that arc length equals radius times angle in radians: s=rθs = r\theta. This formula connects linear distance traveled along a circle to the angular displacement. Let's break this problem into two parts. First, the runner completes 34\frac{3}{4} of a full lap. Since one complete revolution equals 2π2\pi radians, this portion represents 34×2π=3π2\frac{3}{4} \times 2\pi = \frac{3\pi}{2} radians. Next, the runner travels an additional 25π25\pi meters. Using s=rθs = r\theta with s=25πs = 25\pi and r=50r = 50: 25π=50θ25\pi = 50\theta, so θ=25π50=π2\theta = \frac{25\pi}{50} = \frac{\pi}{2} radians. The total angular displacement is 3π2+π2=4π2=2π\frac{3\pi}{2} + \frac{\pi}{2} = \frac{4\pi}{2} = 2\pi radians, confirming answer C. Looking at the wrong answers: A) 7π4\frac{7\pi}{4} likely comes from miscalculating the additional distance as π4\frac{\pi}{4} instead of π2\frac{\pi}{2}. B) 5π2\frac{5\pi}{2} results from incorrectly calculating the additional angle as π\pi radians (perhaps using θ=25π25=π\theta = \frac{25\pi}{25} = \pi). D) 9π4\frac{9\pi}{4} might occur from adding 3π4+6π4\frac{3\pi}{4} + \frac{6\pi}{4}, possibly confusing the radius value in the calculation. Remember: always convert arc length to angle using θ=sr\theta = \frac{s}{r}, and be careful with fraction arithmetic when combining angular displacements.

Question 6

On the unit circle (radius r=1r=1), an angle θ\theta in standard position intercepts an arc that is exactly one quarter of the circle. Based on the information given, what is the radian measure of θ\theta?

  1. π4\frac{\pi}{4}
  2. π2\frac{\pi}{2} (correct answer)
  3. π\pi
  4. 9090
Explanation: This question tests understanding that one quarter of a circle corresponds to π/2 radians. A radian is defined as the measure of an angle that, when placed at the center of a circle, intercepts an arc equal in length to the radius of that circle. Since one complete revolution is 2π radians, one quarter of the circle corresponds to (1/4) × 2π = π/2 radians. Choice B is correct because a quarter circle represents 90° or π/2 radians, which is one-fourth of the full 2π radians. Choice D incorrectly gives the degree measure (90) when the question asks for 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 7

On a unit circle (radius r=1r=1), an angle θ\theta in standard position starts on the positive xx-axis and intercepts an arc of length s=π6s=\frac{\pi}{6}. Based on the information given, what is the radian measure of the angle?

  1. π12\frac{\pi}{12}
  2. π6\frac{\pi}{6} (correct answer)
  3. π3\frac{\pi}{3}
  4. 3030
Explanation: This question tests understanding of radian measure as the arc length on the unit circle. On a unit circle (radius = 1), the radian measure of an angle equals the length of the arc it intercepts. Using the definition θ = s/r, where s = π/6 and r = 1, we find θ = (π/6)/1 = π/6 radians. Choice B is correct because on a unit circle, the numerical value of the angle in radians equals the numerical value of the arc length. Choice D incorrectly gives the answer in degrees (30°) when the question asks for radian measure. 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 8

An angle θ\theta in standard position starts at the positive xx-axis and intercepts an arc of length s=πs=\pi on a circle with radius r=2r=2. Based on the information given, what is the radian measure of θ\theta?

  1. 2π2\pi
  2. π\pi
  3. π2\frac{\pi}{2} (correct answer)
  4. π4\frac{\pi}{4}
Explanation: This question tests understanding of radian measure as the arc length on the unit circle. Radian measure is fundamentally the ratio of arc length to radius: θ = s/r. Using the definition θ = s/r, where s = π and r = 2, we find θ = π/2 radians. Choice C is correct because it uses θ = s/r with s=π and r=2, resulting in π/2 radians. Choice A 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. 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 9

A circle has radius r=10r=10. An angle θ\theta in standard position (starting at the positive xx-axis) intercepts an arc of length s=5πs=5\pi. Using the circle described, what is the radian measure of the angle?

  1. π2\frac{\pi}{2} (correct answer)
  2. 2π2\pi
  3. 5π10\frac{5\pi}{10}
  4. 50π50\pi
Explanation: This question tests understanding of finding radian measure from arc length and radius. Radian measure is fundamentally the ratio of arc length to radius: θ = s/r. Using the definition θ = s/r, where s = 5π and r = 10, we find θ = 5π/10 = π/2 radians. Choice A is correct because it properly applies the fundamental relationship between arc length, radius, and radian measure. Choice D incorrectly multiplies the radius by the arc length (getting 50π) instead of dividing arc length by radius, completely reversing the formula. Remember that radian measure represents how many radius lengths fit into the arc length, so we divide s by r, not multiply.

Question 10

An angle in standard position has its terminal side passing through the point (3,4)(3, 4). If this angle measures α\alpha radians, and another angle measuring α+π2\alpha + \frac{\pi}{2} radians is drawn in standard position, what is the length of the arc intercepted by this second angle on a circle of radius 10?

  1. 10arctan⁡(43)+5π10\arctan\left(\frac{4}{3}\right) + 5\pi (correct answer)
  2. 50arctan⁡(43)+10π50\arctan\left(\frac{4}{3}\right) + 10\pi
  3. 10arctan⁡(34)+5π10\arctan\left(\frac{3}{4}\right) + 5\pi
  4. 5arctan⁡(43)+10π5\arctan\left(\frac{4}{3}\right) + 10\pi
Explanation: First, α=arctan⁡(43)\alpha = \arctan\left(\frac{4}{3}\right) since the terminal side passes through (3,4)(3,4). The second angle is α+π2=arctan⁡(43)+π2\alpha + \frac{\pi}{2} = \arctan\left(\frac{4}{3}\right) + \frac{\pi}{2}. The arc length is s=rθ=10(arctan⁡(43)+π2)=10arctan⁡(43)+5πs = r\theta = 10\left(\arctan\left(\frac{4}{3}\right) + \frac{\pi}{2}\right) = 10\arctan\left(\frac{4}{3}\right) + 5\pi. Choice B multiplies the first term by 5 instead of 10. Choice C uses the reciprocal ratio. Choice D has the coefficients reversed.

Question 11

On a circle with radius r=3r=3, an angle θ\theta is in standard position starting from the positive xx-axis and intercepts an arc of length s=πs=\pi. According to the information given, what is the radian measure of the angle?

  1. 3π3\pi
  2. π3\frac{\pi}{3} (correct answer)
  3. π6\frac{\pi}{6}
  4. 180∘180^\circ
Explanation: This question tests understanding of radian measure as the arc length on the unit circle. Radian measure is fundamentally the ratio of arc length to radius: θ = s/r. Using the definition θ = s/r, where s = π and r = 3, we find θ = π/3 radians. Choice B is correct because it connects to the stimulus data and shows correct application of θ = s/r with s = π and r = 3, resulting in π/3. Choice D 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 12

On the unit circle (radius r=1r=1), an angle θ\theta in standard position starts at the positive xx-axis and intercepts an arc of length s=2s=2. Based on the information given, what is the radian measure of θ\theta?

  1. 22 (correct answer)
  2. 12\frac{1}{2}
  3. 2π2\pi
  4. 2π\frac{2}{\pi}
Explanation: This question tests understanding of radian measure as the arc length on the unit circle. On a unit circle (radius = 1), the radian measure of an angle equals the length of the arc it intercepts. Using the definition θ = s/r, where s = 2 and r = 1, we find θ = 2/1 = 2 radians. Choice A is correct because it connects to the stimulus data and shows correct application of θ = s/r with s = 2 and r = 1. Choice C 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.

Question 13

Using the circle described: a unit circle (radius r=1r=1) with an angle θ\theta in standard position starting from the positive xx-axis. If θ=7π4\theta=\frac{7\pi}{4} radians, what is the length of the intercepted arc ss?

  1. 7π8\frac{7\pi}{8}
  2. 7π4\frac{7\pi}{4} (correct answer)
  3. 9π4\frac{9\pi}{4}
  4. 7π2\frac{7\pi}{2}
Explanation: This question tests understanding of radian measure as the arc length on the unit circle. On a unit circle (radius = 1), the radian measure of an angle equals the length of the arc it intercepts. Using the formula s = rθ, where r = 1 and θ = 7π/4 radians, we calculate s = 1*(7π/4) = 7π/4. Choice B is correct because it applies s = rθ with r=1, yielding the arc length equal to the radian measure 7π/4. Choice A incorrectly halves the numerator, perhaps confusing it with a different fraction. 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. When working with radians, express answers in terms of π rather than decimal approximations unless the context specifically requires decimals.

Question 14

An angle θ\theta is drawn in standard position starting from the positive xx-axis on a unit circle (radius r=1r=1). The intercepted arc length is s=πs=\pi. Based on the information given, which statement correctly describes the relationship between θ\theta and ss on the unit circle?

  1. θ=s2\theta = \frac{s}{2} because the diameter is 22
  2. θ=s\theta = s because r=1r=1 on the unit circle (correct answer)
  3. θ=1s\theta = \frac{1}{s} because angles are reciprocals of arc length
  4. θ=180πs\theta = \frac{180}{\pi}s because radians must be converted to degrees
Explanation: This question tests understanding of radian measure as the arc length on the unit circle. On a unit circle (radius = 1), the radian measure of an angle equals the length of the arc it intercepts. 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 B is correct because it connects to the stimulus data and shows correct application of s = rθ or θ = s/r with r = 1 and s = π, so θ = π. Choice D applies the degree-to-radian conversion factor π/180 when the angle is already in radians. 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.

Question 15

On a unit circle (radius r=1r=1), an angle θ\theta in standard position starts on the positive xx-axis and measures θ=3π2\theta=\frac{3\pi}{2} radians. Using the circle described, what is the length ss of the arc intercepted by the angle?

  1. 3π4\frac{3\pi}{4}
  2. 3π2\frac{3\pi}{2} (correct answer)
  3. 3π3\pi
  4. 270270
Explanation: This question tests understanding of radian measure as the arc length on the unit circle. The relationship between arc length (s), radius (r), and angle measure in radians (θ) is given by s = rθ. Using the formula s = rθ, where r = 1 and θ = 3π/2 radians, we calculate s = 1 × (3π/2) = 3π/2. Choice B is correct because it represents three-quarters of a complete revolution, giving an arc length of 3π/2 on the unit circle. Choice D incorrectly converts the radian measure to degrees (270°) instead of calculating the arc length. When working with radians, express answers in terms of π rather than decimal approximations unless the context specifically requires decimals.

Question 16

Two concentric circles have radii of 5 units and 8 units respectively. If a central angle intercepts arcs of lengths aa and bb on these circles respectively, and b−a=6πb - a = 6\pi, what is the measure of the central angle in radians?

  1. 3π2\frac{3\pi}{2}
  2. 2π2\pi (correct answer)
  3. 4π3\frac{4\pi}{3}
  4. 5π3\frac{5\pi}{3}
Explanation: Let θ\theta be the central angle in radians. Then a=5θa = 5\theta and b=8θb = 8\theta. From b−a=6πb - a = 6\pi: 8θ−5θ=6π8\theta - 5\theta = 6\pi, so 3θ=6π3\theta = 6\pi and θ=2π\theta = 2\pi. Choice A results from solving 8θ−5θ=9π28\theta - 5\theta = \frac{9\pi}{2}. Choice C comes from using b+a=6πb + a = 6\pi instead of b−ab - a. Choice D results from incorrectly setting up 8θ5θ=6ππ\frac{8\theta}{5\theta} = \frac{6\pi}{\pi}.

Question 17

On the unit circle (radius r=1r=1), two angles θ1\theta_1 and θ2\theta_2 are in standard position starting at the positive xx-axis. Angle θ1=π6\theta_1=\frac{\pi}{6} and angle θ2=π2\theta_2=\frac{\pi}{2}. Compared to the arc intercepted by θ1\theta_1, the arc intercepted by θ2\theta_2 is how many times as long?

  1. 33 (correct answer)
  2. 13\frac{1}{3}
  3. π3\frac{\pi}{3}
  4. 66
Explanation: This question tests understanding of radian measure as the arc length on the unit circle. On a unit circle (radius = 1), the radian measure of an angle equals the length of the arc it intercepts. The arc for θ1 = π/6 is s1 = 1*(π/6) = π/6, and for θ2 = π/2 is s2 = 1*(π/2) = π/2, so the ratio s2/s1 = (π/2)/(π/6) = 3. Choice A is correct because it applies s = rθ for both angles with r = 1 and computes the ratio 3. Choice B reverses the ratio, calculating s1/s2 instead of s2/s1. 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 18

On a unit circle (radius r=1r=1), an angle θ\theta in standard position starts on the positive xx-axis and rotates clockwise, intercepting an arc of length s=π6s=\frac{\pi}{6}. Based on the information given, what is the radian measure of the angle?

  1. π6\frac{\pi}{6}
  2. −π6-\frac{\pi}{6} (correct answer)
  3. −6π-6\pi
  4. 3030
Explanation: This question tests understanding of radian measure for clockwise rotation on the unit circle. A radian is defined as the measure of an angle that, when placed at the center of a circle, intercepts an arc equal in length to the radius of that circle. For clockwise rotation, we assign a negative sign to the angle measure, so with an arc length of π/6 traveled clockwise on a unit circle, θ = -π/6 radians. Choice B is correct because clockwise rotation from the positive x-axis results in negative angle measures in standard position. Choice A incorrectly gives a positive value, ignoring that clockwise rotation produces negative angles in the standard coordinate system. Remember that counterclockwise rotation gives positive angles while clockwise rotation gives negative angles; this sign convention is crucial for properly describing rotational direction.

Question 19

Using the unit circle (radius r=1r=1): an angle θ\theta in standard position starts at the positive xx-axis and intercepts an arc of length s=π4s=\frac{\pi}{4}. Which statement correctly describes the relationship between θ\theta and ss in this situation?

  1. θ=s\theta=s because r=1r=1 (correct answer)
  2. θ=1s\theta=\frac{1}{s} because r=1r=1
  3. θ=2s\theta=2s because the diameter is 22
  4. θ=π180s\theta=\frac{\pi}{180}s because radians convert from degrees
Explanation: This question tests understanding of radian measure as the arc length on the unit circle. On a unit circle (radius = 1), the radian measure of an angle equals the length of the arc it intercepts. 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 A is correct because it states θ = s due to r = 1, which matches s = π/4 implying θ = π/4. Choice D applies the degree-to-radian conversion factor π/180 when the angle is already in radians. 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 20

A gear with radius 8 cm is connected to a gear with radius 12 cm. When the smaller gear rotates through an angle of 5π3\frac{5\pi}{3} radians, both gears move the same linear distance along their circumferences. Through what angle does the larger gear rotate?

  1. 5π4\frac{5\pi}{4} radians
  2. 7π6\frac{7\pi}{6} radians
  3. 15π8\frac{15\pi}{8} radians
  4. 10π9\frac{10\pi}{9} radians (correct answer)
Explanation: When you see connected gears, remember that they move the same linear distance along their circumferences - this is the key constraint that determines their relationship. The linear distance traveled equals radius times angle (in radians): s=rθs = r\theta. For the smaller gear with radius 8 cm rotating through 5π3\frac{5\pi}{3} radians, the linear distance is: s=8×5π3=40π3s = 8 \times \frac{5\pi}{3} = \frac{40\pi}{3} cm Since both gears move this same linear distance, you can find the larger gear's rotation angle. For the larger gear with radius 12 cm: 40π3=12θ\frac{40\pi}{3} = 12\theta θ=40π3÷12=40π36=10π9\theta = \frac{40\pi}{3} \div 12 = \frac{40\pi}{36} = \frac{10\pi}{9} radians This confirms answer D is correct. The wrong answers represent common errors: A) 5π4\frac{5\pi}{4} results from incorrectly using the ratio 812×5π3\frac{8}{12} \times \frac{5\pi}{3} but making arithmetic mistakes. B) 7π6\frac{7\pi}{6} comes from confused ratio calculations. C) 15π8\frac{15\pi}{8} occurs when students mistakenly multiply instead of using the inverse relationship - they might think the larger gear rotates through a larger angle, when it actually rotates less. Remember: in connected gears, the smaller gear always rotates through a larger angle than the bigger gear for the same linear distance. The relationship is inversely proportional to their radii. Set up the equation using s=rθs = r\theta for both gears with the same ss.