Algebra 3 Quiz: Degree And Radian Measure
12 questions · exam conditions
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Degree And Radian MeasureQuestion 1 of 12

An angle of −11π6-\frac{11\pi}{6} radians is drawn in standard position. Which degree measure names the same angle for 0∘≤θ<360∘0^\circ \le \theta < 360^\circ?

30∘30^\circ
150∘150^\circ
210∘210^\circ
330∘330^\circ
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Algebra 3 Quiz

Algebra 3 Quiz: Degree And Radian Measure

Practice Degree And Radian Measure in Algebra 3 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 Degree And Radian Measure, giving you a quick way to practice the rules, question types, and explanations that matter most for Algebra 3.

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.

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Question 1

An angle of −11π6-\frac{11\pi}{6} radians is drawn in standard position. Which degree measure names the same angle for 0∘≤θ<360∘0^\circ \le \theta < 360^\circ?

  1. 30∘30^\circ (correct answer)
  2. 150∘150^\circ
  3. 210∘210^\circ
  4. 330∘330^\circ
Explanation: When you see a negative radian angle, think coterminal angles: add or subtract full rotations (2π2\pi or 360∘360^\circ) until the angle falls in the range 0∘≤θ<360∘0^\circ \le \theta < 360^\circ. Here, −11π6-\frac{11\pi}{6} is just short of a full negative rotation, so add one full rotation: −11π6+2π=−11π6+12π6=π6=30∘.-\frac{11\pi}{6} + 2\pi = -\frac{11\pi}{6} + \frac{12\pi}{6} = \frac{\pi}{6} = 30^\circ. So the angle that names the same position is 30∘30^\circ. The other choices are traps from common mistakes. 330∘330^\circ is the positive coterminal angle for −π6-\frac{\pi}{6} or −30∘-30^\circ, not for −11π6-\frac{11\pi}{6}. 150∘150^\circ uses the reference angle in Quadrant II, confusing 180∘−30∘180^\circ-30^\circ with a coterminal angle. 210∘210^\circ is 180∘+30∘180^\circ+30^\circ, placing the angle in Quadrant III, which is a different standard position. None of these differ from 30∘30^\circ by a full rotation. A reliable strategy: convert the radian angle to degrees first if that feels easier, then keep adding or subtracting 360∘360^\circ until it lies in the required range. For any angle, coterminal angles always differ by full circles, not by reference-angle moves.

Question 2

An irrigation sprinkler sprays water in a circular sector. If the spray reaches 24 feet and the sprinkler rotates through 75∘75^\circ, what area is watered in one pass?

  1. 60π60\pi square feet
  2. 120π120\pi square feet (correct answer)
  3. 240π240\pi square feet
  4. 480π480\pi square feet
Explanation: Whenever you see a sprinkler or "circular sector" question, you are finding the area of a slice of a circle. The spray reaches 24 ft, so the radius is 24; the rotation angle is 75°. Multiply the full-circle area by the fraction of the rotation: A=75360π(242)=524π(576)=120πA=\frac{75}{360}\pi(24^2)=\frac{5}{24}\pi(576)=120\pi square feet. The wrong choices are common formula slips. 60π60\pi often comes from converting 75° incorrectly as 75π/360=5π/2475\pi/360=5\pi/24 and then using A=12r2θA=\frac12r^2\theta: 12(576)(5π24)=60π\frac12(576)\left(\frac{5\pi}{24}\right)=60\pi. But the correct radian conversion is 75π/180=5π/1275\pi/180=5\pi/12. 240π240\pi is what you get if you use A=r2θA=r^2\theta and forget the 12\frac12: 576⋅5π12=240π576\cdot\frac{5\pi}{12}=240\pi. 480π480\pi comes from mistakenly using 48 ft, the diameter, as the radius: 75360π(482)=480π\frac{75}{360}\pi(48^2)=480\pi. The radius is 24, not the diameter. Strategy: For sector area, first identify the radius and the angle. Then use A=θ360πr2A=\frac{\theta}{360}\pi r^2 with degrees, or convert to radians and use A=12r2θA=\frac12r^2\theta. Always double-check that you used the radius, not the diameter, and that you included the 12\frac12 in the radian formula.

Question 3

A Ferris wheel has a radius of 20 feet and completes one full turn in 45 seconds. How far does a rider on the rim travel in 15 seconds?

  1. 20π3\frac{20\pi}{3} feet
  2. 40π3\frac{40\pi}{3} feet (correct answer)
  3. 20π20\pi feet
  4. 80π3\frac{80\pi}{3} feet
Explanation: Whenever you see a Ferris wheel or any circular motion problem, you're really dealing with arc length: distance traveled equals the radius times the angle swept out. The wheel completes one full turn, 2π2\pi radians, in 45 seconds. In 15 seconds it sweeps 1545⋅2π=2π3\frac{15}{45}\cdot 2\pi = \frac{2\pi}{3} radians. Multiply by the radius 20 feet: 20⋅2π3=40π320\cdot \frac{2\pi}{3} = \frac{40\pi}{3} feet. That's the distance. Why are the others off? The choice 20π3\frac{20\pi}{3} treats 15 seconds as one-sixth of a turn, using π3\frac{\pi}{3} radians instead of 2π3\frac{2\pi}{3}. The choice 20π20\pi uses half the full circumference of 40π40\pi, which would be the distance for 22.5 seconds, not 15. The choice 80π3\frac{80\pi}{3} is double the correct distance — that's the distance for 30 seconds, or a 23\frac{2}{3} rotation, so it comes from multiplying by an extra factor of 2. For any circular motion question, start by finding the fraction of a full turn, then use arc length s=rθs = r\theta. And check against the full circumference: since the full circle is 40π40\pi feet, one-third of it should be about 13.3π13.3\pi feet. If your answer looks too big or too small, revisit your fraction.

Question 4

A sector has area 18π18\pi square units and a central angle of π4\frac{\pi}{4} radians. What is its radius?

  1. 626\sqrt{2} units
  2. 2424 units
  3. 1212 units (correct answer)
  4. 144144 units
Explanation: This question tests whether you know the sector-area formula and can solve it carefully. A sector is just a "slice" of a circle, so its area is proportional to the central angle. Whenever you see sector area and an angle, reach for A=12r2θA=\frac12 r^2\theta, where θ\theta is in radians. Here, A=18πA=18\pi and θ=π4\theta=\frac{\pi}{4}. Substitute: 18π=12r2(π4)=πr2818\pi=\frac12 r^2\left(\frac{\pi}{4}\right)=\frac{\pi r^2}{8} Cancel π\pi from both sides: 18=r2818=\frac{r^2}{8} Multiply by 8: r2=144r^2=144 So r=12r=12 units. The choice 626\sqrt{2} comes from dropping the 12\frac12 factor: solving 18π=r2⋅π418\pi=r^2\cdot \frac{\pi}{4} gives r2=72r^2=72, which is a classic sign that you forgot the half in the formula. The choice 2424 is simply twice the correct radius — a trap for confusing diameter with radius. The choice 144144 is the value of r2r^2 before you take the square root, so it answers the wrong question. A useful study habit: write the formula as A=12r2θA=\frac12 r^2\theta every time, then check whether your final number is the radius or the radius squared. Taking the square root is the last step — don't skip it.

Question 5

On a circle of radius rr, a central angle intercepts an arc whose length is 2.4r2.4r. What is the angle in degrees?

  1. 2.4∘2.4^\circ
  2. (216π)∘\left(\frac{216}{\pi}\right)^\circ
  3. 432∘432^\circ
  4. (432π)∘\left(\frac{432}{\pi}\right)^\circ (correct answer)
Explanation: When you see arc length and radius together, remember the arc length formula: s=rθs = r\theta, where θ\theta is in radians. Here s=2.4rs = 2.4r, so 2.4r=rθ2.4r = r\theta, giving θ=2.4\theta = 2.4 radians. To convert radians to degrees, multiply by 180π\frac{180}{\pi}: 2.4⋅180π=432π2.4 \cdot \frac{180}{\pi} = \frac{432}{\pi} degrees. That is the correct angle. Now for the traps. Choosing 2.4° treats the radian measure as if it were already degrees; the problem asks for degrees, so conversion is required. Choosing 432° comes from multiplying by 180 instead of by 180π\frac{180}{\pi}, forgetting to divide by π\pi. Choosing 216π∘\frac{216}{\pi}^\circ is half the correct value, likely because the arc length was divided by the diameter 2r2r instead of the radius, giving 1.2 radians instead of 2.4. The key habit is to always keep units straight: the arc length formula gives the angle in radians, and the degree conversion factor is 180π\frac{180}{\pi}, not 180. If you remember that, these circle-angle questions become straightforward. On the exam, whenever a central angle is paired with an arc length and radius, immediately write s=rθs = r\theta, solve for θ\theta, then convert to degrees if needed.

Question 6

Two meshed gears have radii 2 cm and 6 cm. The smaller gear rotates through 180∘180^\circ. Assuming no slipping, through what angle does the larger gear rotate?

  1. π3\frac{\pi}{3} radians (correct answer)
  2. 2π3\frac{2\pi}{3} radians
  3. π\pi radians
  4. 3π3\pi radians
Explanation: Whenever you see meshed gears, a belt, or a pulley, think about the contact point: with no slipping, the arc length traveled along each rim must be equal. Here the smaller gear has radius 2 cm and rotates through 180°, which is π radians. Its rim travels s=rθ=2πs = r\theta = 2\pi cm. The larger gear, radius 6 cm, must travel the same 2π2\pi cm, so 6θ=2π6\theta = 2\pi, giving θ=π3\theta = \frac{\pi}{3} radians. In degrees, that's 60° — the larger gear rotates one-third as far as the smaller gear. The other choices come from losing that arc-length connection. π\pi radians just repeats the smaller gear's own 180°, as if both gears rotate the same amount, which ignores their different radii. 3π3\pi radians reverses the relationship: because the larger gear is three times as big, some people multiply by 3, but the larger gear actually rotates less by that same factor. 2π3\frac{2\pi}{3} radians (120°) often arises from thinking the 1:3 radius ratio means the larger gear completes one-third of a full revolution; in reality, it completes one-third of the small gear's actual angle, (1/3)⋅π=π/3(1/3)\cdot\pi = \pi/3. Study tip: memorize the single relation r1θ1=r2θ2r_1\theta_1 = r_2\theta_2 for any no-slip rotating contact. Convert angles to radians, plug in, and solve — this will protect you from every gear-ratio trap.

Question 7

During a test, the tip of a 20-meter wind-turbine blade moves through a 72∘72^\circ angle. How far does the tip travel?

  1. 4π4\pi meters
  2. 8π8\pi meters (correct answer)
  3. 80π80\pi meters
  4. 14401440 meters
Explanation: When you see a wind-turbine blade sweeping through an angle, think of the blade as the radius of a circle. The distance the tip travels is an arc length, so you need the formula for arc length: s=rθs = r\theta, where θ\theta is in radians, or equivalently, take the fraction of the full circumference. The blade is 20 meters long, so the full circumference of its circular path is 2π(20)=40π2\pi(20) = 40\pi meters. A 72∘72^\circ angle is exactly one-fifth of a full circle, because 72/360=1/572/360 = 1/5. So the tip travels one-fifth of the circumference: 15(40π)=8π\frac{1}{5}(40\pi) = 8\pi meters. That is the correct distance. Now, what about the other choices? The 4π4\pi meters result comes from using πr\pi r instead of 2πr2\pi r for the circumference — a common slip that drops the factor of 2. The 80π80\pi meters is actually the area of the sector swept out by the blade, 15π(20)2\frac{1}{5}\pi(20)^2; it answers a different question, "how much area was swept," not "how far did the tip travel." The 14401440 meters comes from simply multiplying 20 by 72, which ignores the units: degrees are not meters, and you must convert the angle or use a fraction of the circle. Your takeaway: for arc length, always ask whether you are working with distance along the circle or area inside the sector. Convert degrees to a fraction of 360∘360^\circ, then multiply by the full circumference. And sanity-check: the tip can never travel more than the circumference, about 126 meters, so 1440 meters should immediately look impossible.

Question 8

An object moves along a circular path of radius 18 cm and covers an arc length of 12π12\pi cm. Through what angle, in degrees, does it move?

  1. 60∘60^\circ
  2. 120∘120^\circ (correct answer)
  3. 240∘240^\circ
  4. 270∘270^\circ
Explanation: Whenever you see a question involving arc length and radius on a circle, your first thought should be the arc length formula s=rθs = r\theta, where θ\theta is in radians. Here s=12πs = 12\pi cm and r=18r = 18 cm, so solve for θ\theta: θ=sr=12π18=2π3\theta = \frac{s}{r} = \frac{12\pi}{18} = \frac{2\pi}{3} radians. To convert radians to degrees, multiply by 180∘π\frac{180^\circ}{\pi}: 2π3⋅180∘π=120∘.\frac{2\pi}{3} \cdot \frac{180^\circ}{\pi} = 120^\circ. That matches the choice 120∘120^\circ. The other choices represent common missteps. 60∘60^\circ corresponds to π/3\pi/3 radians, which would give arc length 18⋅π3=6π18 \cdot \frac{\pi}{3} = 6\pi cm — exactly half of the given arc. This often happens if you accidentally use the diameter (36 cm) instead of the radius. 240∘240^\circ corresponds to 4π/34\pi/3 radians and would require an arc length of 24π24\pi cm; it may come from doubling the arc or using a radius of 9 cm. 270∘270^\circ is 3π/23\pi/2 radians, giving an arc length of 27π27\pi cm, so it is far too large — a sign that the fraction of the circle was miscalculated. Study tip: after computing, do a sanity check. The circumference is 2π(18)=36π2\pi(18)=36\pi cm. Since 12π12\pi is one-third of that, the angle must be one-third of 360∘360^\circ, or 120∘120^\circ. This verifies your answer quickly.

Question 9

Which of the following degree-radian equivalences is NOT correct?

  1. π3\frac{\pi}{3} radians = 60∘60^\circ
  2. 5π4\frac{5\pi}{4} radians = 225∘225^\circ
  3. 7π6\frac{7\pi}{6} radians = 150∘150^\circ (correct answer)
  4. 11π6\frac{11\pi}{6} radians = 330∘330^\circ
Explanation: Whenever you see a degree-radian equivalence, anchor yourself with the key fact: π\pi radians = 180∘180^\circ. To check any radian measure, multiply by 180π\frac{180}{\pi}, or use common unit-circle equivalents. Here, the equivalence that is NOT correct is 7π6\frac{7\pi}{6} radians = 150∘150^\circ. Converting gives 7π6⋅180π=210∘\frac{7\pi}{6} \cdot \frac{180}{\pi} = 210^\circ, not 150∘150^\circ. This error probably comes from confusing 7π6\frac{7\pi}{6} with 5π6\frac{5\pi}{6}, which is indeed 150∘150^\circ. The other choices are all correct. π3\frac{\pi}{3} radians equals 60∘60^\circ because it is one-third of 180∘180^\circ. 5π4\frac{5\pi}{4} radians equals 225∘225^\circ because 5×45∘=225∘5 \times 45^\circ = 225^\circ, using the fact that π4=45∘\frac{\pi}{4}=45^\circ. And 11π6\frac{11\pi}{6} radians equals 330∘330^\circ because each π6\frac{\pi}{6} step is 30∘30^\circ, so 11×30∘=330∘11 \times 30^\circ = 330^\circ. A fast strategy for radian-degree questions: recognize that π6=30∘\frac{\pi}{6} = 30^\circ and π4=45∘\frac{\pi}{4} = 45^\circ, then multiply the numerator accordingly. Since 7π6\frac{7\pi}{6} means seven 30∘30^\circ slices, it must be 210∘210^\circ. Always pause when a radian measure is near a whole unit-circle landmark—it's easy to swap 150∘150^\circ and 210∘210^\circ on the exam.

Question 10

On a circle with circumference 48π48\pi cm, a central angle of 5π6\frac{5\pi}{6} radians intercepts an arc. What is the arc's length?

  1. 10π10\pi cm
  2. 24π24\pi cm
  3. 20π20\pi cm (correct answer)
  4. 40π40\pi cm
Explanation: When you see a question about arc length, remember that the arc length is a fraction of the circumference determined by the central angle. Specifically, for an angle measured in radians, the arc length is simply the radius times the angle, or equivalently, the circumference times angle2π\frac{\text{angle}}{2\pi}. Here, the circumference is 48π48\pi cm and the central angle is 5π6\frac{5\pi}{6} radians. The arc length is that fraction of the full circle: Arc length=48π×5π62π=48π×512=20π.\text{Arc length} = 48\pi \times \frac{\frac{5\pi}{6}}{2\pi} = 48\pi \times \frac{5}{12} = 20\pi. So the correct answer is 20π20\pi cm. Why the others are off:
  • 10π10\pi cm results from using 5π12\frac{5\pi}{12} (half the angle) or dividing incorrectly — it's half of the true answer.
  • 24π24\pi cm comes from thinking the angle 5π6\frac{5\pi}{6} is 512\frac{5}{12} of a full circle but then multiplying by 22 (or confusing arc length with half the circumference).
  • 40π40\pi cm likely comes from multiplying by 56\frac{5}{6} instead of 512\frac{5}{12} — forgetting to divide by 2π2\pi when using radians.
Your quick check: the full circumference is 48π48\pi. Since 5π6\frac{5\pi}{6} is less than π\pi (half a circle), the arc should be less than half the circumference (less than 24π24\pi). That eliminates 24π24\pi and 40π40\pi, leaving 10π10\pi and 20π20\pi. Knowing the fraction 512\frac{5}{12} gives 20π20\pi. Always reduce the angle fraction before multiplying — it saves time and avoids the common trap of skipping the 2π2\pi denominator.

Question 11

A sector has a radius of 8 centimeters and an arc length of 10π10\pi centimeters. What is the area of the sector?

  1. 80π80\pi square centimeters
  2. 160π160\pi square centimeters
  3. 20π20\pi square centimeters
  4. 40π40\pi square centimeters (correct answer)
Explanation: Whenever you see a sector question involving radius and arc length, connect the sector area formula to the arc length formula: A=12r2θA=\frac12 r^2\theta and s=rθs=r\theta. Here, the arc length is 10π10\pi and radius is 88, so the central angle is θ=sr=10π8=5π4\theta=\frac{s}{r}=\frac{10\pi}{8}=\frac{5\pi}{4} radians. Then A=12(8)2(5π4)=12(64)(5π4)=40πA=\frac12(8)^2\left(\frac{5\pi}{4}\right)=\frac12(64)\left(\frac{5\pi}{4}\right)=40\pi So the sector's area is 40π40\pi square centimeters. The 80π80\pi choice comes from dropping the12\frac12: A=rs=(8)(10π)=80πA=rs=(8)(10\pi)=80\pi. The 160π160\pi choice compounds that error by using the diameter 1616 instead of the radius: 16(10π)=160π16(10\pi)=160\pi. The 20π20\pi choice comes from finding the angle by dividing arc length by diameter instead of radius, giving θ=10π16=5π8\theta=\frac{10\pi}{16}=\frac{5\pi}{8}, then A=12(8)2(5π8)=20πA=\frac12(8)^2(\frac{5\pi}{8})=20\pi. The radius is the correct length to use; save the diameter for when you actually need it. Strategy tip: whenever you know arc length and radius, use A=12rsA=\frac12 r s directly. It is quick and naturally reminds you to include the half.

Question 12

A sector has a radius of 8 centimeters and a central angle measuring 135∘135^\circ. What is the perimeter of the sector?

  1. 6π6\pi cm
  2. 16+24π16+24\pi cm
  3. 16+12π16+12\pi cm
  4. 16+6π16+6\pi cm (correct answer)
Explanation: When you see a sector-perimeter question, remember that a sector is like a pizza slice: its boundary consists of two straight edges, the radii, and one curved edge, the arc. So the perimeter is 2r+arc length2r + \text{arc length}. Here r=8r=8, so the two radii contribute 1616 cm. The central angle is 135∘135^\circ, and arc length is a fraction of the circumference: arc=135360⋅2π(8)\text{arc} = \frac{135}{360}\cdot 2\pi(8). Since 135360=38\frac{135}{360}=\frac38, this becomes 38⋅16π=6π\frac38 \cdot 16\pi = 6\pi. Adding the radii gives 16+6π16+6\pi cm. The trap choices reveal common slips. 6π6\pi cm is just the arc length — it leaves off the two radii entirely. 16+12π16+12\pi cm comes from using 135180\frac{135}{180} instead of 135360\frac{135}{360}, treating the angle as 34\frac34 of a semicircle; the correct fraction is 38\frac38 of a full circle. 16+24π16+24\pi cm adds 1616 to 24π24\pi, which is actually the sector's area (38π(8)2=24π\frac38 \pi(8)^2=24\pi) — mixing up the arc-length formula 2πr2\pi r with the area formula πr2\pi r^2. On exam day, always write down the perimeter components before calculating: two radii plus arc length. Then double-check that your angle fraction is out of 360∘360^\circ, not 180∘180^\circ, and that you used 2πr2\pi r, not πr2\pi r^2. This one formula discipline prevents most sector mistakes.