Circle Area & Circumference - Math 1

1. A quarter-circle sector is cut from a circular sheet of metal with radius 16 inches. What is the perimeter of the remaining piece?
   Answer: $32 + 24\pi$ inches When you see a problem about cutting shapes from circles, you need to carefully track what remains and calculate its complete perimeter. Starting with a circle of radius 16 inches, removing a quarter-circle sector leaves you with three-quarters of the original circle. The perimeter of this remaining piece consists of two parts: the curved edge (three-quarters of the original circumference) plus the two straight edges created by the cuts. For the curved portion: The original circumference is $2\pi r = 2\pi(16) = 32\pi$ inches. Three-quarters of this is $\frac{3}{4} \times 32\pi = 24\pi$ inches. For the straight edges: When you cut out a quarter-circle sector, you make two cuts along radii from the center to the edge. Each cut creates a straight edge equal to the radius length (16 inches). So you have two straight edges of 16 inches each, totaling $16 + 16 = 32$ inches. Therefore, the total perimeter is $32 + 24\pi$ inches. Looking at the wrong answers: Choice A ($64 + 24\pi$) doubles the straight edge length, perhaps from incorrectly thinking you need four radius lengths instead of two. Choice B ($32 + 8\pi$) correctly finds the straight edges but miscalculates the curved portion as one-quarter instead of three-quarters of the circumference. Choice D ($16 + 24\pi$) gets the curved part right but only counts one straight edge instead of two. Remember: when a sector is removed, the remaining perimeter includes both the leftover arc and all newly created straight edges.
2. Three circles with radius 4 cm are arranged so each circle touches the other two at exactly one point. What is the area of the triangular region formed by connecting the centers of the three circles?
   Answer: $16\sqrt{3}$ square cm When three circles of equal radius are arranged so each touches the other two exactly once, you're dealing with externally tangent circles. The key insight is recognizing that the centers of these circles form an equilateral triangle. Since each circle has radius 4 cm and they're externally tangent, the distance between any two centers equals the sum of their radii: 4 + 4 = 8 cm. Because all three distances are equal (8 cm each), the triangle formed by connecting the centers is equilateral with side length 8 cm. To find the area of an equilateral triangle with side length $s$, use the formula: $A = \frac{s^2\sqrt{3}}{4}$. Substituting $s = 8$: $A = \frac{8^2\sqrt{3}}{4} = \frac{64\sqrt{3}}{4} = 16\sqrt{3}$ square cm. Looking at the wrong answers: Choice A ($32\sqrt{3}$) results from incorrectly using the diameter (8) as the radius in the area formula, giving $\frac{16^2\sqrt{3}}{4}$. Choice C ($64\sqrt{3}$) comes from forgetting to divide by 4 in the formula, calculating just $8^2\sqrt{3}$. Choice D ($8\sqrt{3}$) occurs when you mistakenly use the radius (4) instead of the distance between centers (8) as the triangle's side length. The correct answer is B. Remember: when circles are externally tangent, the distance between centers equals the sum of their radii. Always verify whether you need radius or diameter for each part of your calculation.
3. A circular pond has a radius of 15 feet. A concrete walkway 3 feet wide surrounds the pond. What is the area of just the walkway?
   Answer: $99\pi$ square feet The outer radius is 15 + 3 = 18 feet. Walkway area = π(18)² - π(15)² = 324π - 225π = 99π square feet. Choice A represents an incorrect calculation. Choice B gives the total outer area. Choice D gives just the pond area.
4. Two identical circular coins are placed so they touch at exactly one point. If the distance between their centers is 3 cm, what is the total area covered by both coins?
   Answer: $4.5\pi$ square cm When you see two circles touching at exactly one point, you're dealing with externally tangent circles. The key insight is that when two identical circles are externally tangent, the distance between their centers equals twice the radius of each circle. Since the distance between centers is 3 cm, and this equals 2r (where r is the radius of each coin), you can solve: 2r = 3, so r = 1.5 cm. The area of a circle is $\pi r^2$, so each coin has area $\pi(1.5)^2 = 2.25\pi$ square cm. Since the coins only touch at one point (they don't overlap), the total area is simply the sum of both areas: $2.25\pi + 2.25\pi = 4.5\pi$ square cm. Let's examine why the other answers are incorrect: Answer A ($9\pi$) represents the area of a single circle with radius 3 cm, suggesting someone mistakenly used the center-to-center distance as the radius. Answer B ($2.25\pi$) is the area of just one coin, not both. This error occurs when you correctly find the radius but forget to account for having two coins. Answer D ($18\pi$) appears to come from incorrectly calculating $(3)^2 \times 2\pi = 18\pi$, mixing up the area formula or using the wrong radius. Remember: when circles are externally tangent, the center-to-center distance always equals the sum of their radii. For identical circles, this means the distance equals twice one radius. Always double-check whether you need the area of one circle or multiple circles.
5. A bicycle wheel has a diameter of 26 inches. If the wheel makes exactly 50 complete rotations, approximately how far has the bicycle traveled?
   Answer: 4084 inches When you encounter problems about wheels, bicycles, or any circular objects that roll, you're dealing with circumference and distance relationships. The key insight is that when a wheel makes one complete rotation, the bicycle moves forward a distance equal to the wheel's circumference. First, calculate the wheel's circumference using the formula $C = \pi \times d$, where $d$ is the diameter. With a 26-inch diameter: $C = \pi \times 26 \approx 3.14159 \times 26 \approx 81.68$ inches per rotation. For 50 complete rotations, multiply the circumference by the number of rotations: $81.68 \times 50 = 4084$ inches. This matches answer choice D. Let's examine why the other answers are incorrect. Choice A (1300 inches) likely comes from multiplying the diameter by 50 rotations ($26 \times 50 = 1300$), which ignores the circular nature of the wheel entirely. Choice C (2600 inches) results from multiplying diameter by 2, then by 50 ($26 \times 2 \times 50$) - this gets you the radius times 100, but still misses the $\pi$ factor. Choice B (8168 inches) appears to double the correct answer, possibly from a calculation error or confusion about the relationship between circumference and distance. Remember this pattern: distance traveled equals circumference times number of rotations. Don't get trapped into using diameter or radius directly - you must convert to circumference first using $\pi$.
6. A circular fountain has radius 8 feet. Water sprays from the center and lands uniformly in a ring-shaped region between 6 feet and 10 feet from the center. What percentage of the fountain's area receives water?
   Answer: 43.75% The fountain has radius 8 feet, so its total area is $\pi(8)^2 = 64\pi$ square feet. The water lands in a ring from 6 to 10 feet from center, but only the portion within the fountain (6 to 8 feet) receives water. This area is $\pi(8)^2 - \pi(6)^2 = 64\pi - 36\pi = 28\pi$ square feet. The percentage is $\frac{28\pi}{64\pi} \times 100\% = \frac{28}{64} \times 100\% = 43.75\%$. Choice A uses $\frac{16}{64} = 25\%$. Choice B uses $\frac{36}{64} = 56.25\%$. Choice D uses $\frac{48}{64} = 75\%$.
7. A circular dartboard has three concentric regions: a center circle with radius 2 inches, a ring from radius 2 to radius 5 inches, and an outer ring from radius 5 to radius 8 inches. What fraction of the dartboard's area is the middle ring?
   Answer: $\frac{21}{64}$ When you encounter area problems involving concentric circles, you need to find the area of each region by subtracting the areas of the inner circles from the outer circles. To find the middle ring's area, calculate the area of the circle with radius 5 inches, then subtract the area of the inner circle with radius 2 inches. Using the formula $A = \pi r^2$: - Area of circle with radius 5: $\pi(5^2) = 25\pi$ - Area of circle with radius 2: $\pi(2^2) = 4\pi$ - Middle ring area: $25\pi - 4\pi = 21\pi$ The total dartboard area is $\pi(8^2) = 64\pi$ (using the outermost radius of 8 inches). Therefore, the fraction is $\frac{21\pi}{64\pi} = \frac{21}{64}$, which is answer B. Let's examine the wrong answers: A) $\frac{3}{8}$ likely comes from incorrectly using the difference in radii (5-2=3) over the outer radius (8), ignoring that areas involve squared terms. C) $\frac{25}{64}$ results from forgetting to subtract the inner circle's area—this gives you the fraction for everything inside radius 5, not just the ring. D) $\frac{5}{16}$ might come from various calculation errors or misapplying the radius measurements. Study tip: In concentric circle problems, always remember that ring areas require subtraction of the inner circle's area from the outer circle's area. The key insight is that you're finding the area "between" two circles, not just the area of one circle.
8. The circumference of a circular swimming pool is $48\pi$ feet. If a safety rope is installed 3 feet from the edge of the pool all around, what is the total area enclosed by the safety rope?
   Answer: $729\pi$ square feet From the pool's circumference $C = 2\pi r = 48\pi$, we get $r = 24$ feet. The safety rope is 3 feet from the pool edge, so it forms a circle with radius $24 + 3 = 27$ feet. The area enclosed by the safety rope is $A = \pi(27)^2 = 729\pi$ square feet. Choice A uses $(24)^2 + 3^2 = 576 + 72 = 648$ incorrectly. Choice C uses only the pool's area $\pi(24)^2 = 576\pi$. Choice D uses $(24 + 3)^2 - 24 = 729 - 24 = 675$ incorrectly.
9. A circular pizza is cut into 8 equal slices. If each slice has an area of $4.5\pi$ square inches, what is the diameter of the pizza?
   Answer: $12$ inches Total pizza area = 8 × 4.5π = 36π square inches. Since πr² = 36π, we have r² = 36, so r = 6 inches. Therefore, diameter = 2r = 12 inches. Choice B gives the radius instead of diameter. Choice C doubles the correct answer. Choice D uses an incorrect area calculation (assuming 9π per slice instead of 4.5π).
10. The area of a circular track is $324\pi$ square meters. A runner completes exactly 5 laps around the track. What total distance does the runner cover?
    Answer: $180\pi$ meters When you encounter problems involving circular tracks, you need to connect area and circumference through the radius. The key insight is that area helps you find the radius, which then determines the circumference (distance around the track). Start with the area formula: $A = \pi r^2$. Since the area is $324\pi$, you have $324\pi = \pi r^2$. Dividing both sides by $\pi$ gives $r^2 = 324$, so $r = 18$ meters. Now find the circumference (distance of one lap): $C = 2\pi r = 2\pi(18) = 36\pi$ meters. Since the runner completes 5 laps, the total distance is $5 \times 36\pi = 180\pi$ meters. Looking at the wrong answers: Choice A ($1620\pi$) appears to come from multiplying the area by 5 instead of finding the circumference first—this confuses area with distance. Choice B ($36\pi$) gives you the circumference of just one lap, forgetting to multiply by 5 laps. Choice C ($90\pi$) might result from calculation errors, possibly using $r = 9$ instead of $r = 18$, then multiplying by 5. The correct answer is D: $180\pi$ meters. Remember this sequence for circular track problems: area → radius → circumference → total distance. Don't confuse area (space enclosed) with circumference (distance around). Always verify your radius calculation before proceeding, and don't forget to multiply by the number of laps at the end.
11. A circular garden bed is surrounded by a brick border that is 2 feet wide. If the total area (garden bed plus border) is $144\pi$ square feet and the garden bed alone has area $100\pi$ square feet, what is the circumference of the outer edge of the border?
    Answer: $24\pi$ feet This problem involves concentric circles - a garden bed surrounded by a border. When you encounter circular geometry problems with borders or rings, always identify the inner and outer radii first. Since the garden bed has area $100\pi$ square feet, you can find its radius using $A = \pi r^2$. Setting $100\pi = \pi r^2$, you get $r^2 = 100$, so the inner radius is $r = 10$ feet. The total area (garden plus border) is $144\pi$ square feet. Using the same formula: $144\pi = \pi R^2$, where $R$ is the outer radius. This gives $R^2 = 144$, so $R = 12$ feet. You can verify this makes sense: the border width should be $12 - 10 = 2$ feet, which matches the given information. The circumference of the outer edge is $C = 2\pi R = 2\pi(12) = 24\pi$ feet, confirming answer D. Let's examine the wrong answers: Answer A ($20\pi$) represents the circumference of the inner circle: $2\pi(10) = 20\pi$. This is a common trap - using the garden's circumference instead of the border's outer edge. Answer B ($28\pi$) might result from incorrectly adding the border width to the circumference rather than the radius. Answer C ($22\pi$) could come from using an average radius of 11 feet, which isn't mathematically meaningful here. Remember: in concentric circle problems, always work with areas first to find radii, then calculate the requested measurement. Don't confuse inner and outer boundaries.
12. A semicircular window has a perimeter of $20 + 10\pi$ feet. What is the area of the window?
    Answer: $50\pi$ square feet The perimeter of a semicircle consists of the diameter plus half the circumference: diameter + πr = 2r + πr = r(2 + π). So r(2 + π) = 20 + 10π, giving r = 10. The area of a semicircle is (1/2)πr² = (1/2)π(10)² = 50π. Choice B forgets the factor of 1/2. Choice C uses the full circle area. Choice D incorrectly doubles the calculation.
13. A circular field has an area of $441\pi$ square meters. A farmer walks around the perimeter of the field twice, then walks directly across the field through its center. What is the total distance walked?
    Answer: $84\pi + 42$ meters From πr² = 441π, we get r² = 441, so r = 21 meters. Circumference = 2π(21) = 42π meters. Walking around twice = 2 × 42π = 84π meters. Walking across the center (diameter) = 2r = 42 meters. Total = 84π + 42 meters. Choice A converts everything to decimal incorrectly. Choice B uses only one lap around the perimeter. Choice C uses one lap and radius instead of diameter.
14. A sector of a circle has a central angle of $120°$ and an area of $12\pi$ square inches. What is the circumference of the complete circle?
    Answer: $12\pi$ inches A sector with central angle $120° = \frac{120°}{360°} = \frac{1}{3}$ of the circle has area $\frac{1}{3}\pi r^2 = 12\pi$. Solving: $\pi r^2 = 36\pi$, so $r^2 = 36$ and $r = 6$ inches. The circumference of the complete circle is $C = 2\pi r = 2\pi(6) = 12\pi$ inches. Choice B uses $3 \times 6\pi = 18\pi$. Choice C uses $4 \times 6\pi = 24\pi$. Choice D uses the full circle area instead of circumference.
15. A circular pizza is cut into 8 equal slices. If each slice has an area of $4.5\pi$ square inches, what is the diameter of the pizza?
    Answer: 12 inches If each slice has area $4.5\pi$ square inches and there are 8 slices, the total pizza area is $8 \times 4.5\pi = 36\pi$ square inches. Using $A = \pi r^2$: $36\pi = \pi r^2$, so $r^2 = 36$ and $r = 6$ inches. Therefore, the diameter is $2r = 12$ inches. Choice A gives the radius instead of diameter. Choice B incorrectly uses $4.5 \times 2 = 9$. Choice D incorrectly uses $36$ without taking the square root.
16. A sector of a circle has a central angle of $120°$ and an area of $12\pi$ square inches. What is the circumference of the entire circle?
    Answer: $12\pi$ inches When you encounter sector problems, remember that sectors are proportional parts of a circle. The key relationship is that a sector's area relates to the total circle's area the same way the central angle relates to $360°$. Let's work backwards from the given information. The sector has a central angle of $120°$ and an area of $12\pi$ square inches. Since $120°$ is $\frac{120°}{360°} = \frac{1}{3}$ of the full circle, this sector represents one-third of the entire circle's area. If $\frac{1}{3}$ of the circle has area $12\pi$, then the full circle's area is $12\pi \times 3 = 36\pi$ square inches. Now we can find the radius using the circle area formula: $A = \pi r^2$. So $36\pi = \pi r^2$, which gives us $r^2 = 36$, therefore $r = 6$ inches. The circumference is $C = 2\pi r = 2\pi(6) = 12\pi$ inches, which is choice B. Choice A ($36\pi$) incorrectly uses the total area as the circumference. Choice C ($6\pi$) represents $\pi r$ instead of the full circumference formula $2\pi r$. Choice D ($24\pi$) likely comes from doubling the sector area incorrectly or miscalculating the proportional relationship. Study tip: In sector problems, always identify what fraction of the circle you're dealing with first (angle ÷ $360°$), then use proportions to find the missing information. Keep the area formula ($\pi r^2$) and circumference formula ($2\pi r$) separate to avoid confusion.
17. Two circles have the same center. The larger circle has an area that is 4 times the area of the smaller circle. If the circumference of the smaller circle is $12\pi$ units, what is the circumference of the larger circle?
    Answer: $24\pi$ units From the smaller circle's circumference 12π = 2πr, we get r = 6. Its area is π(6)² = 36π. The larger circle has area 4 × 36π = 144π. So πR² = 144π, giving R² = 144, so R = 12. The larger circumference is 2π(12) = 24π. Choice B (48π) assumes the circumference scales by the area ratio. Choice C (16π) uses an incorrect radius calculation. Choice D (36π) assumes circumference scales by 3 when area scales by 4.

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