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ACT Math

ACT Math Help: Circles

Review real example questions for Circles in ACT Math.

Question 1

What is the area of a circle with diameter 101010 inches?​​​

  1. 25π25\pi25π
  2. 100π100\pi100π
  3. 50π50\pi50π
  4. 10π10\pi10π
Explanation: We need to find the area of a circle with diameter 10 inches. The area formula is A = πr², and since diameter = 10, the radius = 5. Substituting: A = π(5)² = 25π square inches. Choice B incorrectly uses the diameter (10) instead of the radius (5) in the formula, giving π(10)² = 100π.

Question 2

What is the area of sector with central angle 120° in a circle with radius 8?

  1. 128π3\frac{128\pi}{3}3128π​
  2. 32π3\frac{32\pi}{3}332π​
  3. 64π6\frac{64\pi}{6}664π​
  4. 64π3\frac{64\pi}{3}364π​
Explanation: We need to find the sector area with central angle 120° in a circle with radius 8. The sector area formula is sector = (θ/360°) × πr². Substituting: sector = (120°/360°) × π(8)² = (1/3) × 64π = 64π/3. Choice B incorrectly uses 180° instead of 360°, while choice A doubles the correct result.

Question 3

On a coordinate plane, a circle has equation (x+1)2+(y−4)2=64(x+1)^2+(y-4)^2=64(x+1)2+(y−4)2=64. What is the radius of the circle?

  1. 444
  2. 646464
  3. 161616
  4. 888
Explanation: We are finding the radius of a circle given by the equation (x + 1)² + (y - 4)² = 64. The standard form is (x - h)² + (y - k)² = r², so r = √(right-hand side). Here, r = √64 = 8. This matches choice D. Choice B incorrectly uses r² = 64 as the radius, and choice C might double it thinking of diameter, while choice A halves the square root erroneously.

Question 4

Find the area of a circle with radius 7.

  1. 28π28\pi28π
  2. 14π14\pi14π
  3. 21π21\pi21π
  4. 49π49\pi49π
Explanation: We need to find the area of a circle with radius 7. The area formula is A=πr2A = \pi r^2A=πr2. Substituting r = 7: A=π(7)2=49πA = \pi (7)^2 = 49\piA=π(7)2=49π. Choice B incorrectly uses the circumference formula 2πr2\pi r2πr, while choice C uses an incorrect coefficient.

Question 5

What is the area of a circle with radius 12?

  1. 24π24\pi24π
  2. 72π72\pi72π
  3. 144π144\pi144π
  4. 36π36\pi36π
Explanation: We need to find the area of a circle with radius 12. The area formula is A = πr². Substituting r = 12: A = π(12)² = 144π. Choice B (72π) uses the circumference formula 2πr instead of area, while choice A (24π) uses just 2πr, and choice D (36π) uses an incorrect calculation.

Question 6

If the circumference of a circle is 20π20\pi20π, what is its radius?

  1. 5
  2. 10
  3. 20
  4. 15
Explanation: We need to find the radius when the circumference is 20π. The circumference formula is C = 2πr, so 20π = 2πr. Dividing both sides by 2π: r = 20π/(2π) = 10. Choice A (5) would give a circumference of 10π, while choice C (20) would give a circumference of 40π.

Question 7

What is the radius of a circle if the circumference is 16π16\pi16π?

  1. 4
  2. 8
  3. 16
  4. 32
Explanation: We need to find the radius when the circumference is 16π. The circumference formula is C = 2πr, so 16π = 2πr. Dividing both sides by 2π: r = 16π/(2π) = 8. Choice A (4) would give a circumference of 8π, while choice C (16) would give a circumference of 32π.

Question 8

A circle has an area of 64π64\pi64π. What is its diameter?

  1. 8
  2. 16
  3. 32
  4. 12
Explanation: We need to find the diameter when area is 64π. Using A = πr², we have 64π = πr², so r² = 64, giving r = 8. The diameter is 2r = 2(8) = 16. Choice A uses only the radius, while choice C doubles the area instead of finding the diameter.

Question 9

A circle has radius 555. What is the area of the circle?

  1. 10π10\pi10π
  2. 25π25\pi25π
  3. 5π5\pi5π
  4. 50π50\pi50π
Explanation: We need to find the area of a circle with radius 5. The area formula is A = πr². Substituting r = 5: A = π(5)² = 25π. Choice A incorrectly uses the circumference formula 2πr = 10π, while choice C gives only the radius value.

Question 10

A circle has a diameter of 20. What is the area of the circle?

  1. 100π100\pi100π
  2. 400π400\pi400π
  3. 200π200\pi200π
  4. 300π300\pi300π
Explanation: We need to find the area with diameter 20. Since diameter = 20, radius r = 10. Using the area formula A = πr²: A = π(10)² = 100π. Choice B incorrectly uses the diameter squared, choice C uses diameter times π, and choice D uses an arbitrary coefficient.