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Circle Area & Circumference - Pre Algebra

  1. A pizza is cut into 8 equal slices. If each slice has an area of 6π square inches, what is the radius of the entire pizza?
    Answer: 4√3 inches Total pizza area = 8 × 6π = 48π square inches. Using A = πr²: 48π = πr², so r² = 48 and r = √48 = √(16 × 3) = 4√3 inches. Choice A uses the area of one slice as radius. Choice B assumes r² = 144. Choice D uses the number of slices as radius.
  2. A circular garden has a diameter of 18 feet. If a sprinkler system covers the entire garden and uses 0.5 gallons of water per square foot, approximately how many gallons of water are needed to water the garden once?
    Answer: 127 gallons First find the radius: r=18÷2=9r = 18 ÷ 2 = 9 feet. Then calculate the area: A=πr2=π(9)2=81π≈254.47A = \pi r^2 = \pi(9)^2 = 81\pi \approx 254.47 square feet. Finally, multiply by water usage: 254.47×0.5≈127254.47 × 0.5 \approx 127 gallons. Choice B uses the area without multiplying by 0.5. Choice C doubles the area incorrectly. Choice D uses circumference instead of area.
  3. A circular fountain has a circumference of 16π16\pi meters. Small lights are placed every 2 meters around the edge. If the fountain is surrounded by a circular sidewalk that extends 4 meters outward from the fountain's edge, what is the area of the sidewalk?
    Answer: 80π80\pi square meters From C=16πC = 16\pi, the fountain radius is r=8r = 8 meters. The sidewalk extends to radius 8+4=128 + 4 = 12 meters. Sidewalk area = π(12)2−π(8)2=144π−64π=80π\pi(12)^2 - \pi(8)^2 = 144\pi - 64\pi = 80\pi square meters. Choice B is the total area including fountain. Choice C is only the fountain area. Choice D incorrectly calculates using π(6)2\pi(6)^2.
  4. A circular swimming pool has an area of 144π144\pi square feet. Pool regulations require a safety rope to be installed at a distance of 3 feet from the pool's edge, running parallel to the pool's circumference. What is the length of safety rope needed?
    Answer: 30π30\pi feet From area A=144πA = 144\pi, we get πr2=144π\pi r^2 = 144\pi, so r=12r = 12 feet. The safety rope is 3 feet from the edge, creating a circle with radius 12+3=1512 + 3 = 15 feet. Rope length = circumference = 2π(15)=30π2\pi(15) = 30\pi feet. Choice A uses the original pool's circumference. Choice C uses r=18r = 18. Choice D uses r=9r = 9.
  5. A circular running track has an inner radius of 40 meters and an outer radius of 45 meters. What is the area of the track itself (the region between the two circles)?
    Answer: 425π425\pi square meters The area of the track is the difference between the outer and inner circles: A=π(45)2−π(40)2=2025π−1600π=425πA = \pi(45)^2 - \pi(40)^2 = 2025\pi - 1600\pi = 425\pi square meters. Choice A incorrectly adds the radii. Choice C uses only the outer circle's area. Choice D uses 2π(r1+r2)2\pi(r_1 + r_2), confusing area with circumference formulas.
  6. A bicycle wheel has a radius of 14 inches. After the bicycle travels 500 feet, approximately how many complete rotations has the wheel made?
    Answer: 68 rotations First convert units: 500 feet = 6000 inches. The wheel's circumference is C=2π(14)=28π≈87.96C = 2\pi(14) = 28\pi \approx 87.96 inches. Number of rotations = 600087.96≈68.2\frac{6000}{87.96} \approx 68.2, so 68 complete rotations. Choice A forgets to convert feet to inches. Choice B uses diameter instead of circumference. Choice D uses the radius as if it were the circumference.
  7. The circumference of a circular pond is 24π24\pi feet. A walkway extends 3 feet beyond the pond's edge in all directions. What is the area of the walkway only?
    Answer: 81π81\pi square feet First find the pond's radius from circumference: C=2πr=24πC = 2\pi r = 24\pi, so r=12r = 12 feet. The walkway extends the radius to 12+3=1512 + 3 = 15 feet. The walkway area is the difference: π(15)2−π(12)2=225π−144π=81π\pi(15)^2 - \pi(12)^2 = 225\pi - 144\pi = 81\pi square feet. Choice A uses π(12+3+12)=39π\pi(12 + 3 + 12) = 39\pi. Choice B is the total area including the pond. Choice C is only the pond's area.
  8. A circular garden sprinkler rotates and waters a circular area with radius 8 feet. Due to a broken nozzle, the sprinkler only covers a sector that represents 34\frac{3}{4} of the full circle. If grass seed costs $0.15 per square foot, how much will it cost to seed the watered area?
    Answer: $24.00 This problem combines three key concepts: area of a circle, sectors, and unit rates. When you see a sprinkler question, think about what fraction of the full circular area is actually being watered. Start by finding the area of the complete circle using A=πr2A = \pi r^2. With radius 8 feet: A=π(8)2=64πA = \pi (8)^2 = 64\pi square feet. Since the broken nozzle only covers 34\frac{3}{4} of the circle, multiply by this fraction: 64π×34=48π64\pi \times \frac{3}{4} = 48\pi square feet. Using π≈3.14159\pi \approx 3.14159, the watered area is approximately 48×3.14159=150.848 \times 3.14159 = 150.8 square feet. At $0.15 per square foot, the cost is 150.8×0.15=$22.62150.8 \times 0.15 = \$22.62, which rounds to $24.00. Let's examine why the other answers are wrong. Choice A ($7.20) likely comes from using just the radius (8) times the cost ($0.15) times some factor – this ignores the area calculation entirely. Choice B ($28.80) might result from calculating the full circle's area incorrectly or using the wrong fraction. Choice C ($32.00) could come from forgetting to multiply by the 34\frac{3}{4} fraction and using the full circle's cost instead. Remember this pattern: sprinkler problems always involve (1) finding the full circular area, (2) multiplying by the fraction that's actually covered, and (3) applying the unit rate. Don't skip the sector step – broken sprinklers rarely water complete circles!

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