Degree & Radian Measure - Algebra 3

1. An arc is three times as long as its radius. What is the central angle in degrees?
   Answer: $\left(\frac{540}{\pi}\right)^\circ$ If the arc length is $s = 3r$, then $\theta = \frac{s}{r} = 3$ radians. Converting to degrees gives $3\cdot \frac{180^\circ}{\pi} = \frac{540^\circ}{\pi} \approx 171.9^\circ$. The $540^\circ$ choice results from not dividing by $\pi$, $\frac{180}{\pi}$ is one radian, and $3^\circ$ treats radians as if they were degrees.
2. A sector has arc length $12$ cm and area $18$ cm$^2$. Which of the following gives its radius and central angle?
   Answer: radius $3$ cm; central angle $4$ rad Since $A = \frac{1}{2}rs$, we have $18 = \frac{1}{2}r(12) = 6r$, so $r = 3$. Then $\theta = \frac{s}{r} = \frac{12}{3} = 4$ rad. The $r = \frac{3}{2}$, $\theta = 8$ pair results from omitting the factor $\frac{1}{2}$ in the area formula; the other pairs satisfy $r\theta = 12$ but do not give the stated area.
3. During a test, the tip of a 20-meter wind-turbine blade moves through a $72^\circ$ angle. How far does the tip travel?
   Answer: $8\pi$ meters 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: $s = r\theta$, where $\theta$ 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 $2\pi(20) = 40\pi$ meters. A $72^\circ$ angle is exactly one-fifth of a full circle, because $72/360 = 1/5$. So the tip travels one-fifth of the circumference: $\frac{1}{5}(40\pi) = 8\pi$ meters. That is the correct distance. Now, what about the other choices? The $4\pi$ meters result comes from using $\pi r$ instead of $2\pi r$ for the circumference — a common slip that drops the factor of 2. The $80\pi$ meters is actually the area of the sector swept out by the blade, $\frac{1}{5}\pi(20)^2$; it answers a different question, "how much area was swept," not "how far did the tip travel." The $1440$ 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 $360^\circ$, 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.
4. Which of the following degree-radian equivalences is NOT correct?
   Answer: $\frac{7\pi}{6}$ radians = $150^\circ$ Whenever you see a degree-radian equivalence, anchor yourself with the key fact: $\pi$ radians = $180^\circ$. To check any radian measure, multiply by $\frac{180}{\pi}$, or use common unit-circle equivalents. Here, the equivalence that is NOT correct is $\frac{7\pi}{6}$ radians = $150^\circ$. Converting gives $\frac{7\pi}{6} \cdot \frac{180}{\pi} = 210^\circ$, not $150^\circ$. This error probably comes from confusing $\frac{7\pi}{6}$ with $\frac{5\pi}{6}$, which is indeed $150^\circ$. The other choices are all correct. $\frac{\pi}{3}$ radians equals $60^\circ$ because it is one-third of $180^\circ$. $\frac{5\pi}{4}$ radians equals $225^\circ$ because $5 \times 45^\circ = 225^\circ$, using the fact that $\frac{\pi}{4}=45^\circ$. And $\frac{11\pi}{6}$ radians equals $330^\circ$ because each $\frac{\pi}{6}$ step is $30^\circ$, so $11 \times 30^\circ = 330^\circ$. A fast strategy for radian-degree questions: recognize that $\frac{\pi}{6} = 30^\circ$ and $\frac{\pi}{4} = 45^\circ$, then multiply the numerator accordingly. Since $\frac{7\pi}{6}$ means seven $30^\circ$ slices, it must be $210^\circ$. Always pause when a radian measure is near a whole unit-circle landmark—it's easy to swap $150^\circ$ and $210^\circ$ on the exam.
5. A sector of a circle has area $10\pi$ and radius $5$. What is its perimeter?
   Answer: $10+4\pi$ From $A = \frac{1}{2}r^2\theta$, solve $10\pi = \frac{25}{2}\theta$, so $\theta = \frac{4\pi}{5}$. The arc length is $5\cdot \frac{4\pi}{5} = 4\pi$, and the perimeter includes two radii: $2(5)+4\pi = 10+4\pi$. The $4\pi$ choice omits the radii, and $10+2\pi$ uses half the correct angle.
6. A sector has a radius of 8 centimeters and an arc length of $10\pi$ centimeters. What is the area of the sector?
   Answer: $40\pi$ square centimeters Whenever you see a sector question involving radius and arc length, connect the sector area formula to the arc length formula: $A=\frac12 r^2\theta$ and $s=r\theta$. Here, the arc length is $10\pi$ and radius is $8$, so the central angle is $\theta=\frac{s}{r}=\frac{10\pi}{8}=\frac{5\pi}{4}$ radians. Then $$A=\frac12(8)^2\left(\frac{5\pi}{4}\right)=\frac12(64)\left(\frac{5\pi}{4}\right)=40\pi$$ So the sector's area is $40\pi$ square centimeters. The $80\pi$ choice comes from dropping the$\frac12$: $A=rs=(8)(10\pi)=80\pi$. The $160\pi$ choice compounds that error by using the diameter $16$ instead of the radius: $16(10\pi)=160\pi$. The $20\pi$ choice comes from finding the angle by dividing arc length by diameter instead of radius, giving $\theta=\frac{10\pi}{16}=\frac{5\pi}{8}$, then $A=\frac12(8)^2(\frac{5\pi}{8})=20\pi$. 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 $A=\frac12 r s$ directly. It is quick and naturally reminds you to include the half.
7. On a circle with circumference $48\pi$ cm, a central angle of $\frac{5\pi}{6}$ radians intercepts an arc. What is the arc's length?
   Answer: $20\pi$ cm 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 $\frac{\text{angle}}{2\pi}$. Here, the circumference is $48\pi$ cm and the central angle is $\frac{5\pi}{6}$ radians. The arc length is that fraction of the full circle: $$\text{Arc length} = 48\pi \times \frac{\frac{5\pi}{6}}{2\pi} = 48\pi \times \frac{5}{12} = 20\pi.$$ So the correct answer is $20\pi$ cm. Why the others are off: - $10\pi$ cm results from using $\frac{5\pi}{12}$ (half the angle) or dividing incorrectly — it's half of the true answer. - $24\pi$ cm comes from thinking the angle $\frac{5\pi}{6}$ is $\frac{5}{12}$ of a full circle but then multiplying by $2$ (or confusing arc length with half the circumference). - $40\pi$ cm likely comes from multiplying by $\frac{5}{6}$ instead of $\frac{5}{12}$ — forgetting to divide by $2\pi$ when using radians. Your quick check: the full circumference is $48\pi$. Since $\frac{5\pi}{6}$ is less than $\pi$ (half a circle), the arc should be less than half the circumference (less than $24\pi$). That eliminates $24\pi$ and $40\pi$, leaving $10\pi$ and $20\pi$. Knowing the fraction $\frac{5}{12}$ gives $20\pi$. Always reduce the angle fraction before multiplying — it saves time and avoids the common trap of skipping the $2\pi$ denominator.
8. A wheel of radius $30$ cm rotates at $240$ revolutions per minute. What is the linear speed of a point on its rim?
   Answer: $240\pi$ cm/s Since $240$ rpm is $4$ rev/s, the angular speed is $\omega = 4\cdot 2\pi = 8\pi$ rad/s. Thus $v = r\omega = 30\cdot 8\pi = 240\pi$ cm/s. The smaller choices correspond to using $60$, $120$, or $180$ rpm instead of $240$ rpm.
9. Two meshed gears have radii 2 cm and 6 cm. The smaller gear rotates through $180^\circ$. Assuming no slipping, through what angle does the larger gear rotate?
   Answer: $\frac{\pi}{3}$ radians 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 $s = r\theta = 2\pi$ cm. The larger gear, radius 6 cm, must travel the same $2\pi$ cm, so $6\theta = 2\pi$, giving $\theta = \frac{\pi}{3}$ 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. $\pi$ radians just repeats the smaller gear's own 180°, as if both gears rotate the same amount, which ignores their different radii. $3\pi$ 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. $\frac{2\pi}{3}$ 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, $(1/3)\cdot\pi = \pi/3$. Study tip: memorize the single relation $r_1\theta_1 = r_2\theta_2$ for any no-slip rotating contact. Convert angles to radians, plug in, and solve — this will protect you from every gear-ratio trap.
10. At 3:30, what is the radian measure of the smaller angle between the hour and minute hands?
    Answer: $\frac{5\pi}{12}$ At 3:30 the minute hand is at $180^\circ$. The hour hand has moved $3.5\cdot 30^\circ = 105^\circ$. The smaller difference is $75^\circ$, and $75^\circ = \frac{75\pi}{180} = \frac{5\pi}{12}$. The other choices come from assuming the hour hand is fixed at 3 or placing it at the 4 o'clock position.
11. An arc of length $10\pi$ lies on a circle of radius $6$. What is the central angle in degrees?
    Answer: $300^\circ$ The central angle in radians is $\theta = \frac{s}{r} = \frac{10\pi}{6} = \frac{5\pi}{3}$. Converting gives $\frac{5\pi}{3}\cdot \frac{180^\circ}{\pi} = 300^\circ$. The $150^\circ$ choice commonly comes from using $2\pi r$ in the denominator; the $60^\circ$ and $240^\circ$ choices come from rotational/arithmetic slips in placing $\frac{5\pi}{3}$.
12. A sector has a radius of 8 centimeters and a central angle measuring $135^\circ$. What is the perimeter of the sector?
    Answer: $16+6\pi$ cm 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 $2r + \text{arc length}$. Here $r=8$, so the two radii contribute $16$ cm. The central angle is $135^\circ$, and arc length is a fraction of the circumference: $\text{arc} = \frac{135}{360}\cdot 2\pi(8)$. Since $\frac{135}{360}=\frac38$, this becomes $\frac38 \cdot 16\pi = 6\pi$. Adding the radii gives $16+6\pi$ cm. The trap choices reveal common slips. $6\pi$ cm is just the arc length — it leaves off the two radii entirely. $16+12\pi$ cm comes from using $\frac{135}{180}$ instead of $\frac{135}{360}$, treating the angle as $\frac34$ of a semicircle; the correct fraction is $\frac38$ of a full circle. $16+24\pi$ cm adds $16$ to $24\pi$, which is actually the sector's area ($\frac38 \pi(8)^2=24\pi$) — mixing up the arc-length formula $2\pi r$ with the area formula $\pi r^2$. 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 $360^\circ$, not $180^\circ$, and that you used $2\pi r$, not $\pi r^2$. This one formula discipline prevents most sector mistakes.
13. On a circle of radius $r$, a central angle intercepts an arc whose length is $2.4r$. What is the angle in degrees?
    Answer: $\left(\frac{432}{\pi}\right)^\circ$ When you see arc length and radius together, remember the arc length formula: $s = r\theta$, where $\theta$ is in radians. Here $s = 2.4r$, so $2.4r = r\theta$, giving $\theta = 2.4$ radians. To convert radians to degrees, multiply by $\frac{180}{\pi}$: $2.4 \cdot \frac{180}{\pi} = \frac{432}{\pi}$ 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 $\frac{180}{\pi}$, forgetting to divide by $\pi$. Choosing **$\frac{216}{\pi}^\circ$** is half the correct value, likely because the arc length was divided by the diameter $2r$ 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 $\frac{180}{\pi}$, 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 $s = r\theta$, solve for $\theta$, then convert to degrees if needed.
14. 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?
    Answer: $\frac{40\pi}{3}$ feet 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, $2\pi$ radians, in 45 seconds. In 15 seconds it sweeps $\frac{15}{45}\cdot 2\pi = \frac{2\pi}{3}$ radians. Multiply by the radius 20 feet: $20\cdot \frac{2\pi}{3} = \frac{40\pi}{3}$ feet. That's the distance. Why are the others off? The choice $\frac{20\pi}{3}$ treats 15 seconds as one-sixth of a turn, using $\frac{\pi}{3}$ radians instead of $\frac{2\pi}{3}$. The choice $20\pi$ uses half the full circumference of $40\pi$, which would be the distance for 22.5 seconds, not 15. The choice $\frac{80\pi}{3}$ is double the correct distance — that's the distance for 30 seconds, or a $\frac{2}{3}$ 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 $s = r\theta$. And check against the full circumference: since the full circle is $40\pi$ feet, one-third of it should be about $13.3\pi$ feet. If your answer looks too big or too small, revisit your fraction.
15. An angle is measured as $135^\circ 36'$. What is its measure in radians?
    Answer: $\frac{113\pi}{150}$ Because $36' = 36/60 = 0.6^\circ$, the angle is $135.6^\circ$. Its radian measure is $135.6\cdot \frac{\pi}{180} = \frac{1356\pi}{1800} = \frac{113\pi}{150}$. The other choices come from misreading the minute portion or from simplifying $135.6/180$ incorrectly.
16. An angle of $-\frac{11\pi}{6}$ radians is drawn in standard position. Which degree measure names the same angle for $0^\circ \le \theta < 360^\circ$?
    Answer: $30^\circ$ When you see a negative radian angle, think coterminal angles: add or subtract full rotations ($2\pi$ or $360^\circ$) until the angle falls in the range $0^\circ \le \theta < 360^\circ$. Here, $-\frac{11\pi}{6}$ is just short of a full negative rotation, so add one full rotation: $$-\frac{11\pi}{6} + 2\pi = -\frac{11\pi}{6} + \frac{12\pi}{6} = \frac{\pi}{6} = 30^\circ.$$ So the angle that names the same position is $30^\circ$. The other choices are traps from common mistakes. $330^\circ$ is the positive coterminal angle for $-\frac{\pi}{6}$ or $-30^\circ$, not for $-\frac{11\pi}{6}$. $150^\circ$ uses the reference angle in Quadrant II, confusing $180^\circ-30^\circ$ with a coterminal angle. $210^\circ$ is $180^\circ+30^\circ$, placing the angle in Quadrant III, which is a different standard position. None of these differ from $30^\circ$ by a full rotation. A reliable strategy: convert the radian angle to degrees first if that feels easier, then keep adding or subtracting $360^\circ$ until it lies in the required range. For any angle, coterminal angles always differ by full circles, not by reference-angle moves.
17. A sector has area $24\pi$ and radius $6$. What is its central angle in radians?
    Answer: $\frac{4\pi}{3}$ Using $A = \frac{1}{2}r^2\theta$, solve $24\pi = \frac{1}{2}(6^2)\theta = 18\theta$, so $\theta = \frac{24\pi}{18} = \frac{4\pi}{3}$. The $\frac{2\pi}{3}$ result comes from omitting the factor $\frac{1}{2}$, and $4\pi$ comes from using $r$ rather than $r^2$ in the formula.
18. An object moves along a circular path of radius 18 cm and covers an arc length of $12\pi$ cm. Through what angle, in degrees, does it move?
    Answer: $120^\circ$ Whenever you see a question involving arc length and radius on a circle, your first thought should be the arc length formula $s = r\theta$, where $\theta$ is in radians. Here $s = 12\pi$ cm and $r = 18$ cm, so solve for $\theta$: $$\theta = \frac{s}{r} = \frac{12\pi}{18} = \frac{2\pi}{3}$$ radians. To convert radians to degrees, multiply by $\frac{180^\circ}{\pi}$: $$\frac{2\pi}{3} \cdot \frac{180^\circ}{\pi} = 120^\circ.$$ That matches the choice $120^\circ$. The other choices represent common missteps. $60^\circ$ corresponds to $\pi/3$ radians, which would give arc length $18 \cdot \frac{\pi}{3} = 6\pi$ cm — exactly half of the given arc. This often happens if you accidentally use the diameter (36 cm) instead of the radius. $240^\circ$ corresponds to $4\pi/3$ radians and would require an arc length of $24\pi$ cm; it may come from doubling the arc or using a radius of 9 cm. $270^\circ$ is $3\pi/2$ radians, giving an arc length of $27\pi$ 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 $2\pi(18)=36\pi$ cm. Since $12\pi$ is one-third of that, the angle must be one-third of $360^\circ$, or $120^\circ$. This verifies your answer quickly.
19. Sector $A$ has radius $6$ and central angle $75^\circ$. Sector $B$ has radius $3$ and has the same area as sector $A$. What is the central angle of sector $B$ in radians?
    Answer: $\frac{5\pi}{3}$ Convert $75^\circ = \frac{5\pi}{12}$. Sector $A$ has area $\frac{1}{2}(6^2)(\frac{5\pi}{12}) = \frac{15\pi}{2}$. For sector $B$, $\frac{1}{2}(3^2)\theta = \frac{15\pi}{2}$, so $\theta = \frac{15\pi}{2} \cdot \frac{2}{9} = \frac{5\pi}{3}$. The other choices come from using an incorrect converted angle for sector $A$, such as $60^\circ$ or $30^\circ$.
20. Which of the following statements is false?
    Answer: A sector of radius $4$ and central angle $72^\circ$ has area $\frac{8\pi}{5}$. Check each: $\frac{7\pi}{10}$ rad $=126^\circ$. Also, $40^\circ = \frac{2\pi}{9}$ rad, so the arc length is $9\cdot \frac{2\pi}{9}=2\pi$. For a $72^\circ = \frac{2\pi}{5}$ sector of radius 4, the area is $\frac{1}{2}(4^2)(\frac{2\pi}{5}) = \frac{16\pi}{5}$, while the arc length is $4\cdot \frac{2\pi}{5} = \frac{8\pi}{5}$. Thus the false statement is the one claiming the area is $\frac{8\pi}{5}$.

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