ACT Math · Learn by Concept

ACT Math Help: Circles

Review real example questions for Circles in ACT Math.

Question 1 / 10

0 of 10 answered

What is the length of arc AB with central angle 120° in a circle with radius 6?

All questions

Question 1

What is the length of arc AB with central angle 120° in a circle with radius 6?

$6\pi$
$8\pi$
$4\pi$ (correct answer)
$12\pi$

Explanation: Arc length is a fraction of the full circumference, and the fraction is the central angle over $360^\circ$ . Here that fraction is $\frac{120^\circ}{360^\circ} = \frac{1}{3}$ , and the full circumference is $2\pi r = 2\pi(6) = 12\pi$ , so the arc measures $\frac{1}{3}(12\pi) = 4\pi$ . The answer $12\pi$ is the entire circumference, which is what you get if you compute $2\pi r$ and forget to take the angle's fraction of it, and $8\pi$ comes from dividing by $180^\circ$ instead of $360^\circ$ , which makes the fraction $\frac{2}{3}$ rather than $\frac{1}{3}$ . Whenever a problem asks for part of a circle, compute the whole quantity first, then multiply by the angle fraction $\frac{\theta}{360^\circ}$ as the last step.

Question 2

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

$\frac{128\pi}{3}$
$\frac{32\pi}{3}$
$\frac{64\pi}{6}$
$\frac{64\pi}{3}$ (correct answer)

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

What is the area of a circle with radius 12?

$24\pi$
$72\pi$
$144\pi$ (correct answer)
$36\pi$

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 4

A circle has an area of $64\pi$ . What is its diameter?

8
16 (correct answer)
32
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 5

A circle is represented by $(x + 1)^2 + (y - 2)^2 = 49$ . What is the radius?

3.5
14
7 (correct answer)
49

Explanation: We need to find the radius from the circle equation (x + 1)² + (y - 2)² = 49. In standard form (x - h)² + (y - k)² = r², we have r² = 49, so r = √49 = 7. The center is (-1, 2), but we only need the radius. Choice B incorrectly uses the diameter.

Question 6

What is the measure of the arc with a central angle of 120° in a circle with radius 3?

$4\pi$
$3\pi$
$6\pi$
$2\pi$ (correct answer)

Explanation: We need the arc length with central angle 120° and radius 3. Using the arc length formula s = (θ/360°) × 2πr: s = (120°/360°) × 2π(3) = (1/3) × 6π = 2π. Choice B uses 180° instead of 120°, while choice C uses the full circumference.

Question 7

A circle has a radius of 6 centimeters. What is the area, in square centimeters, of a sector of this circle that has a central angle of $\dfrac{2\pi}{3}$ radians?

$4\pi$
$12\pi$ (correct answer)
$24\pi$
$36\pi$

Explanation: The correct answer is B (12π). The area of a sector is A = (1/2)r²θ, where θ is in radians. Substitute: A = (1/2)(6²)(2π/3) = (1/2)(36)(2π/3) = (36π/3) = 12π. A (4π) uses the arc length formula (rθ = 6 × 2π/3 = 4π) instead of the area formula. C (24π) applies the sector fraction to the full area but forgets the 1/2 factor: (2π/3)/(2π) × 36π = 12π... actually (2/3) × 36π = 24π, dropping the 1/2. D (36π) computes the full circle area πr² = 36π without applying any sector fraction. Memorize the sector area formula: A = (1/2)r²θ with θ in radians.

Question 8

A circle is given by the equation $(x-4)^2+(y+1)^2=64$ . What is the radius of the circle?

$64$
$8$ (correct answer)
$4$
$16$

Explanation: We need to find the radius from the equation (x - 4)² + (y + 1)² = 64. In the standard form (x - h)² + (y - k)² = r², the right side equals r². Since r² = 64, we have r = √64 = 8. Choice A incorrectly gives 64, which is r² not r, while choice D gives 16, which would be the diameter.

Question 9

Which equation represents a circle with center (7, 8) and radius 9?

$(x - 7)^2 + (y - 8)^2 = 9$
$(x + 7)^2 + (y + 8)^2 = 81$
$(x - 7)^2 + (y - 8)^2 = 81$ (correct answer)
$(x - 7)^2 + (y + 8)^2 = 9$

Explanation: We need the equation of a circle with center (7, 8) and radius 9. Using standard form (x - h)² + (y - k)² = r²: (x - 7)² + (y - 8)² = 9² = 81. Choice A incorrectly uses the radius instead of radius squared, choice B changes the center signs, and choice D changes the y-coordinate sign.

Question 10

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

$4$
$64$
$16$
$8$ (correct answer)

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.