ACT Math Quiz: Circles

What this quiz covers

This quiz focuses on Circles, giving you a quick way to practice the rules, question types, and explanations that matter most for ACT Math.

How to use this quiz

Try each quiz question before looking at the correct answer. Use the explanations to review missed ideas, then come back to similar questions until the pattern feels familiar.

Question 1

1. $6\pi$
2. $8\pi$
3. $4\pi$ (correct answer)
4. $12\pi$

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

1. $\frac{128\pi}{3}$
2. $\frac{32\pi}{3}$
3. $\frac{64\pi}{6}$
4. $\frac{64\pi}{3}$ (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

1. $24\pi$
2. $72\pi$
3. $144\pi$ (correct answer)
4. $36\pi$

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

1. 8
2. 16 (correct answer)
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 5

1. 3.5
2. 14
3. 7 (correct answer)
4. 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

1. $4\pi$
2. $3\pi$
3. $6\pi$
4. $2\pi$ (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

1. $4\pi$
2. $12\pi$ (correct answer)
3. $24\pi$
4. $36\pi$

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

1. $64$
2. $8$ (correct answer)
3. $4$
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

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

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

1. $4$
2. $64$
3. $16$
4. $8$ (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.

Question 11

1. $8\pi$
2. $16\pi$
3. $64\pi$ (correct answer)
4. $256\pi$

Explanation: This is a circles question testing the relationship between circumference and area. Choice C (64π) is correct — find the radius from circumference: C = 2πr = 16π → r = 8. Area = πr² = π(8²) = 64π. Choice A (8π) uses the radius value (r = 8) directly as the area — computing π × r instead of π × r². Forgetting to square the radius is the most common error on this type. Choice B (16π) simply restates the given circumference value as the area, confusing the two circle formulas. Choice D (256π) uses the diameter (d = 16) instead of the radius in the area formula: π(16²) = 256π — a factor-of-4 error since (2r)² = 4r². Pro tip: On any circle problem, extract the radius first — from circumference (r = C/2π), from diameter (r = d/2), or from area (r = √(A/π)). Then plug r into the formula you need. Never plug in the diameter where the radius belongs.

Question 12

1. 5 (correct answer)
2. 10
3. 14
4. 25

Explanation: This is a completing the square question testing the standard form of a circle. Choice A (5) is correct — complete the square for x: (x² − 6x + 9). Complete for y: (y² + 8y + 16). Add the same constants to the right side: 0 + 9 + 16 = 25. Equation becomes: (x − 3)² + (y + 4)² = 25. Since r² = 25, the radius r = 5. Choice B (10) results from adding the half-coefficients used in completing the square (3 + 4 = 7... or (−6/2) + (8/2) = −3 + 4 = 1... actually possibly adding 6 + 4 = 10 as a shortcut). Choice C (14) adds the original x and y coefficients without completing the square: 6 + 8 = 14. Choice D (25) correctly finds r² = 25 but reports r² instead of r — forgetting to take the square root. Pro tip: In completing the square for a circle, take half of each linear coefficient and square it to find what to add: for −6x, add (−6/2)² = 9; for +8y, add (8/2)² = 16. Add these same values to the right side. Then r = √(right side), not the right side itself.

Question 13

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

Explanation: This is an arc length question testing the fraction-of-circumference method. Choice B (2π) is correct — a 60° central angle is 60/360 = 1/6 of the full circle. Circumference = 2πr = 2π(6) = 12π. Arc length = (1/6)(12π) = 2π. Choice A (π) results from finding 1/6 of the radius (6/6 = 1) and using that as the arc, not 1/6 of the circumference. Choice C (3π) results from computing 1/6 of an incorrect circumference, possibly using 18π (computing C = πd = π × 6 instead of π × 12). Choice D (6π) uses the full radius as the arc — not accounting for either the fraction of the circle or the circumference formula. Pro tip: Arc length = (central angle/360°) × 2πr. Always compute the full circumference first, then take your fraction of it. Never use the radius or diameter directly as the arc length.

Question 14

1. 0
2. 1
3. 2 (correct answer)
4. 4

Explanation: This is a circles and intersections question testing substitution into a circle equation. Choice C (2) is correct — substitute x = 0 into x² + y² = 25: 0 + y² = 25 → y² = 25 → y = ±5. Two intersection points: (0, 5) and (0, −5). Choice A (0) suggests no intersections — the student may think x = 0 doesn't cross the circle, or makes an error computing y² = 25. Choice B (1) finds only y = 5, forgetting the negative solution — y² = 25 has two solutions. Choice D (4) confuses the number of intersections with the radius or diameter. Pro tip: To find intersections of a line and a circle, substitute the line's equation into the circle equation. x = 0 is the y-axis. Substituting eliminates x entirely, leaving a simple equation for y. The equation y² = 25 always has two solutions (positive and negative square roots) unless you're looking for a tangent point where the discriminant is zero.

Question 15

1. $7\pi$
2. $14\pi$ (correct answer)
3. $24.5\pi$
4. $98\pi$

Explanation: The correct answer is B (14π). The area formula for a circle is A = πr². Set equal to 49π: πr² = 49π → r² = 49 → r = 7. The circumference is C = 2πr = 2π(7) = 14π. A (7π) uses only r in the circumference formula: C = πr = 7π, forgetting the factor of 2. C (24.5π) divides the area value by 2 instead of finding r first: 49π ÷ 2 = 24.5π. D (98π) treats 49 as the radius instead of r², computing C = 2π(49) = 98π. Always find the radius first from the area, then substitute into the circumference formula.

Question 16

1. $4$ (correct answer)
2. $8$
3. $16$
4. $\sqrt{52}$

Explanation: The correct answer is A (4). Complete the square for both x and y. Group: (x² − 10x) + (y² + 6y) = −18. Complete the square: add (10/2)² = 25 and (6/2)² = 9 to both sides: (x² − 10x + 25) + (y² + 6y + 9) = −18 + 25 + 9 = 16. Rewrite: (x − 5)² + (y + 3)² = 16. This is standard circle form with r² = 16, so r = 4. B (8) doubles the radius, possibly confusing radius with diameter. C (16) reports r² instead of r — forgetting to take the square root. D (√52) results from an arithmetic error in the completing-the-square process. Always complete the square for both variables and take the square root of the right side to get r.

Question 17

1. $98\pi$
2. $49\pi$ (correct answer)
3. $196\pi$
4. $28\pi$

Explanation: We need to find the area of a circle with diameter 14. The area formula is A = πr², and since diameter = 14, the radius = 7. Substituting r = 7: A = π(7)² = 49π. Choice A (98π) doubles the correct result, while choice C (196π) incorrectly uses the diameter squared in the formula.

Question 18

1. $x^2 + y^2 = 100$ (correct answer)
2. $x^2 + y^2 = 10$
3. $(x-10)^2 + y^2 = 100$
4. $(x+10)^2 + y^2 = 10$

Explanation: We need to write the equation of a circle with center (0, 0) and radius 10. The standard form is (x-h)² + (y-k)² = r², where (h,k) is the center. Substituting center (0, 0) and r = 10: (x-0)² + (y-0)² = 10², which simplifies to x² + y² = 100. Choice B incorrectly uses the radius instead of radius squared, while choices C and D unnecessarily shift the center.

Question 19

1. $(x-5)^2+y^2=144$
2. $x^2+(y+5)^2=144$
3. $x^2+(y-5)^2=12$
4. $x^2+(y-5)^2=144$ (correct answer)

Explanation: We need the equation of a circle with center (0, 5) and radius 12. The standard form is (x - h)² + (y - k)² = r² where (h, k) is the center. Substituting: (x - 0)² + (y - 5)² = 12², which gives x² + (y - 5)² = 144. Choice B incorrectly uses (y + 5), while choice A places the center at (5, 0) instead.

Question 20

1. $64\pi$
2. $16\pi$ (correct answer)
3. $8\pi$
4. $128\pi$

Explanation: This question asks for the circumference of the circle given by (x + 1)² + (y - 4)² = 64. The equation is in standard form, so radius r = √64 = 8. The circumference formula is C = 2πr, giving 2π(8) = 16π. Choice A confuses area πr² = 64π with circumference. Always extract the radius from the equation first, then apply the correct formula without mixing area and circumference.