Arc Length & Sector Area - Math 3

1. A sector of a circle has an area of $48\pi$ square units and a central angle of $\frac{2\pi}{3}$ radians. If the radius is increased by $50\%$ while maintaining the same central angle, what is the new arc length?
   Answer: $18\pi$ units First, find the original radius using the sector area formula: $A = \frac{1}{2}r^2\theta$. So $48\pi = \frac{1}{2}r^2 \cdot \frac{2\pi}{3}$, which gives $48\pi = \frac{\pi r^2}{3}$. Solving: $r^2 = 144$, so $r = 12$. The new radius is $12 \times 1.5 = 18$. The new arc length is $s = r\theta = 18 \times \frac{2\pi}{3} = 12\pi \times \frac{3}{2} = 18\pi$. Choice B uses the original radius for arc length. Choice C incorrectly applies the 50% increase as multiplication by 1.5 to the angle instead of radius. Choice D doubles the correct answer by confusing diameter with radius.
2. A pendulum swings through an arc of length $24$ cm. If the pendulum string is $18$ cm long, what is the area of the sector swept by the pendulum?
   Answer: $216$ square cm Given arc length $s = 24$ cm and radius $r = 18$ cm. First find the central angle: $\theta = \frac{s}{r} = \frac{24}{18} = \frac{4}{3}$ radians. Then calculate sector area: $A = \frac{1}{2}r^2\theta = \frac{1}{2}(18)^2 \cdot \frac{4}{3} = \frac{1}{2} \cdot 324 \cdot \frac{4}{3} = 216$ square cm. Choice A uses $A = rs$ incorrectly. Choice C uses $A = r^2\theta$ without the $\frac{1}{2}$ factor. Choice D uses $A = \frac{1}{2}rs$ which is incorrect formula.
3. A sector has central angle $\theta$ radians and radius $r$ units. If both the central angle and radius are doubled, by what factor does the arc length increase, and by what factor does the sector area increase?
   Answer: Arc length increases by factor of 4; area increases by factor of 8 Original arc length: $s_1 = r\theta$. New arc length: $s_2 = (2r)(2\theta) = 4r\theta = 4s_1$. So arc length increases by factor of 4. Original area: $A_1 = \frac{1}{2}r^2\theta$. New area: $A_2 = \frac{1}{2}(2r)^2(2\theta) = \frac{1}{2} \cdot 4r^2 \cdot 2\theta = 8 \cdot \frac{1}{2}r^2\theta = 8A_1$. So area increases by factor of 8. Choice A incorrectly calculates arc length factor. Choice C uses factor 16 for area (which would be if we cubed instead of squared and doubled). Choice D switches the factors incorrectly.
4. Two concentric circles have radii of $8$ units and $12$ units. A sector is formed between two radii with a central angle of $\frac{2\pi}{3}$ radians. What is the area of the region between the two circular arcs?
   Answer: $\frac{80\pi}{3}$ square units When you encounter concentric circles with a sector, you're finding the area between two circular arcs. This requires calculating two sector areas and finding their difference. The area of a sector is given by $A = \frac{1}{2}r^2\theta$, where $r$ is the radius and $\theta$ is the central angle in radians. For the larger circle (radius = 12): $A_{large} = \frac{1}{2}(12)^2 \cdot \frac{2\pi}{3} = \frac{1}{2} \cdot 144 \cdot \frac{2\pi}{3} = \frac{288\pi}{6} = \frac{48\pi}{1} = 48\pi$ For the smaller circle (radius = 8): $A_{small} = \frac{1}{2}(8)^2 \cdot \frac{2\pi}{3} = \frac{1}{2} \cdot 64 \cdot \frac{2\pi}{3} = \frac{128\pi}{6} = \frac{64\pi}{3}$ The area between the arcs is: $48\pi - \frac{64\pi}{3} = \frac{144\pi}{3} - \frac{64\pi}{3} = \frac{80\pi}{3}$ This confirms answer C is correct. Answer A ($\frac{96\pi}{3}$) likely comes from incorrectly adding the two sector areas instead of subtracting them. Answer B ($\frac{64\pi}{3}$) is just the area of the smaller sector alone, missing the subtraction step entirely. Answer D ($\frac{56\pi}{3}$) suggests a calculation error, possibly from using incorrect radius values or making arithmetic mistakes. Study tip: For concentric circle problems, always subtract the inner area from the outer area. Double-check your sector formula: it's $\frac{1}{2}r^2\theta$, not $\frac{1}{2}r\theta$ or other variations.
5. A pendulum swings through an arc length of $24$ cm. If the pendulum arm is $40$ cm long and the sector area swept is $480$ square cm, what is the central angle in radians?
   Answer: $0.6$ radians Using the arc length formula: $s = r\theta$, so $24 = 40\theta$, giving $\theta = 0.6$ radians. We can verify using the sector area formula: $A = \frac{1}{2}r^2\theta = \frac{1}{2}(40)^2(0.6) = \frac{1}{2}(1600)(0.6) = 480$ square cm, which matches. Choice B uses $\theta = \frac{s}{r} \times \frac{4}{3}$. Choice C incorrectly uses $\theta = \frac{2s}{r}$. Choice D uses $\theta = \frac{s}{2r}$.
6. Two identical sectors are placed together to form a larger sector. Each small sector has radius $8$ units and central angle $\frac{\pi}{6}$ radians. If they are arranged so their central angles add up, what is the arc length of the combined sector?
   Answer: $\frac{8\pi}{3}$ units When you encounter problems involving sectors and arc length, remember that arc length depends on both the radius and the central angle. The formula is: arc length = radius × central angle (in radians). Here, you have two identical sectors, each with radius 8 units and central angle $\frac{\pi}{6}$ radians. When placed together so their central angles add up, the combined sector has the same radius (8 units) but a central angle of $\frac{\pi}{6} + \frac{\pi}{6} = \frac{\pi}{3}$ radians. Using the arc length formula: arc length = $8 \times \frac{\pi}{3} = \frac{8\pi}{3}$ units. This confirms answer B is correct. Now let's examine why the other answers are wrong. Answer A gives $\frac{4\pi}{3}$, which you'd get if you mistakenly used radius 4 instead of 8, or if you forgot to add the central angles together. Answer C is identical to A, suggesting this is a common error the test makers expect. Answer D gives $\frac{8\pi}{6}$, which simplifies to $\frac{4\pi}{3}$ – this occurs when you multiply the radius by each individual angle ($8 \times \frac{\pi}{6}$) rather than by the combined angle. Key strategy: In sector problems, always identify what's changing and what stays constant. When sectors are combined, the radius typically stays the same while angles add up. Double-check that you're using the total central angle, not the individual angles, in your arc length calculation.
7. A circular pizza is cut into sectors. One slice has a central angle of $\frac{\pi}{4}$ radians and an area of $18\pi$ square inches. If another slice from the same pizza has an arc length of $6\pi$ inches, what is its area?
   Answer: $36\pi$ square inches When you encounter problems about pizza slices or circular sectors, you're working with proportional relationships between central angles, arc lengths, and areas. The key insight is that all these measurements are proportional to each other for sectors of the same circle. First, let's find the pizza's radius using the given slice. For a sector with central angle $\theta$ and radius $r$, the area formula is $A = \frac{1}{2}r^2\theta$. With the first slice having angle $\frac{\pi}{4}$ and area $18\pi$: $18\pi = \frac{1}{2}r^2 \cdot \frac{\pi}{4}$ Solving: $18\pi = \frac{\pi r^2}{8}$, so $r^2 = 144$ and $r = 12$ inches. Now for the second slice with arc length $6\pi$ inches. Since arc length $s = r\theta$, we have $6\pi = 12\theta$, giving us $\theta = \frac{\pi}{2}$ radians. The area of this slice is: $A = \frac{1}{2}(12)^2 \cdot \frac{\pi}{2} = \frac{144\pi}{4} = 36\pi$ square inches. Choice A ($30\pi$) might result from calculation errors in finding the radius. Choice B ($24\pi$) could come from incorrectly using the first slice's area formula. Choice C ($48\pi$) likely stems from using $A = r^2\theta$ instead of $A = \frac{1}{2}r^2\theta$. Remember: always find the radius first when working with sectors, then use it consistently in your formulas. The factor of $\frac{1}{2}$ in the sector area formula is crucial—don't forget it.
8. Two sectors of different circles have the same arc length of $12\pi$ units. The first sector has radius $9$ units and the second has radius $16$ units. What is the difference between their areas?
   Answer: $42\pi$ square units First find the central angles using $s = r\theta$. For first sector: $\theta_1 = \frac{12\pi}{9} = \frac{4\pi}{3}$ radians. For second sector: $\theta_2 = \frac{12\pi}{16} = \frac{3\pi}{4}$ radians. Now find areas: $A_1 = \frac{1}{2}(9)^2 \cdot \frac{4\pi}{3} = \frac{81 \cdot 4\pi}{6} = \frac{324\pi}{6} = 54\pi$. $A_2 = \frac{1}{2}(16)^2 \cdot \frac{3\pi}{4} = \frac{256 \cdot 3\pi}{8} = \frac{768\pi}{8} = 96\pi$. Difference = $96\pi - 54\pi = 42\pi$ square units. Choice B gives the smaller area alone. Choice C uses arithmetic error. Choice D uses wrong angle calculations.
9. A sector has a central angle of $\frac{3\pi}{4}$ radians. If the ratio of the sector's arc length to its radius is $k$, what is the ratio of the sector's area to the square of its radius?
   Answer: $\frac{k}{2}$ When you encounter sector problems, remember that sectors are defined by their central angle, and all sector formulas depend on this angle measured in radians. Given information: central angle = $\frac{3\pi}{4}$ radians, and the ratio of arc length to radius is $k$. Let's call the radius $r$. The arc length formula is $s = r\theta$, where $\theta$ is the central angle. So the arc length is $r \cdot \frac{3\pi}{4} = \frac{3\pi r}{4}$. Since we're told that $\frac{\text{arc length}}{\text{radius}} = k$, we have $k = \frac{3\pi r/4}{r} = \frac{3\pi}{4}$. The sector area formula is $A = \frac{1}{2}r^2\theta$. Substituting our angle: $A = \frac{1}{2}r^2 \cdot \frac{3\pi}{4} = \frac{3\pi r^2}{8}$. The ratio of sector area to the square of radius is $\frac{A}{r^2} = \frac{3\pi r^2/8}{r^2} = \frac{3\pi}{8}$. Since $k = \frac{3\pi}{4}$, we can write this as $\frac{k}{2}$. Choice A gives $\frac{3\pi}{8}$, which equals our answer numerically but misses that the problem asks for the answer in terms of $k$. Choice C squares $k$ incorrectly—there's no reason to square the arc-to-radius ratio. Choice D multiplies by $\frac{3}{8}$ instead of $\frac{1}{2}$, confusing the sector area coefficient with the final ratio. Remember: sector area is $\frac{1}{2}r^2\theta$, and when problems give you ratios involving $k$, express your final answer in terms of $k$ rather than the underlying angle.
10. A sector has an arc length that is $\frac{3}{4}$ of the radius and a sector area of $54$ square units. What is the radius of the circle?
    Answer: $12$ units When you encounter sector problems, you need to connect arc length and area using the fundamental sector formulas. Both quantities depend on the radius and central angle, so you can use the given relationships to solve for the radius. Start with the arc length formula: $s = r\theta$, where $s$ is arc length, $r$ is radius, and $\theta$ is the central angle in radians. Since the arc length is $\frac{3}{4}$ of the radius, you have $\frac{3r}{4} = r\theta$. Dividing both sides by $r$ gives you $\theta = \frac{3}{4}$ radians. Now use the sector area formula: $A = \frac{1}{2}r^2\theta$. Substituting the known values: $54 = \frac{1}{2}r^2 \cdot \frac{3}{4}$. Simplifying: $54 = \frac{3r^2}{8}$. Multiply both sides by $\frac{8}{3}$: $r^2 = 54 \cdot \frac{8}{3} = 144$. Therefore, $r = 12$ units. Let's check why the other answers are wrong. Choice A (16 units): This would give $\theta = \frac{3}{16}$ and an area of $\frac{1}{2}(256)(\frac{3}{16}) = 24$, not 54. Choice B (9 units): This gives $\theta = \frac{1}{3}$ and an area of $\frac{1}{2}(81)(\frac{1}{3}) = 13.5$, far too small. Choice D (8 units): This produces $\theta = \frac{3}{8}$ and an area of $\frac{1}{2}(64)(\frac{3}{8}) = 12$, also too small. Remember: when dealing with sectors, always find the central angle first using the simpler relationship, then substitute into the more complex formula to solve for the unknown.

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