Circles - ACT Math

1. Which equation represents a circle with center (3, -2) and radius 5?
   Answer: $(x-3)^2 + (y+2)^2 = 25$ We need to write the equation of a circle with center (3, -2) and radius 5. The standard form is (x-h)² + (y-k)² = r², where (h,k) is the center. Substituting center (3, -2) and r = 5: (x-3)² + (y-(-2))² = 5², which simplifies to (x-3)² + (y+2)² = 25. Choice B incorrectly places the center at (-3, 2) instead of (3, -2).
2. A circle has area $49\pi$. What is the radius of the circle?
   Answer: $7$ This question seeks the radius of a circle with area 49π. The area formula is A = πr², so solve for r by r² = A/π = 49π/π = 49, then r = √49 = 7. Choice A incorrectly assumes r² = 49 without dividing by π. Choice D halves 49 unnecessarily. Always isolate r by dividing by π first and taking the square root carefully.
3. The area of a circle is $81\pi$ square units. What is the radius of the circle?
   Answer: $9$ We are finding the radius of a circle with area 81π square units. The correct formula for area is A = πr², so solving for r gives r = √(A/π). Substituting, r = √(81π/π) = √81 = 9 units. This matches choice B. Choice A incorrectly sets r² = 81π instead of 81, and choice C uses the diameter as 18, confusing radius with diameter.
4. A circle has circumference $18\pi$. What is the diameter of the circle?
   Answer: $18$ We need to find the diameter when circumference is 18π. Using the circumference formula C = πd, we have 18π = πd. Dividing both sides by π: d = 18π/π = 18. Choice A gives the radius instead of diameter, while choice C incorrectly doubles the diameter.
5. What is the area of sector with central angle 120° in a circle with radius 8?
   Answer: $\frac{64\pi}{3}$ 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.
6. A circle has circumference $30\pi$ cm. What is the radius of the circle?
   Answer: $15$ This question asks for the radius of a circle with a circumference of $30\pi$ cm. The correct formula for circumference is $C = 2\pi r$, so solve for r by $r = C / (2\pi)$. Substituting gives $r = 30\pi / (2\pi) = 30 / 2 = 15$ cm. Make sure to divide by $2\pi$ fully to isolate the radius. Choice A, 30, forgets to divide by 2, and choice C, $30/\pi$, divides only by \pi instead of 2\pi.
7. Which equation represents a circle with center (-4, 5) and radius 6?
   Answer: $(x+4)^2 + (y-5)^2 = 36$ We need to write the equation of a circle with center (-4, 5) and radius 6. The standard form is (x-h)² + (y-k)² = r², where (h,k) is the center. Substituting center (-4, 5) and r = 6: (x-(-4))² + (y-5)² = 6², which simplifies to (x+4)² + (y-5)² = 36. Choice B incorrectly places the center at (4, -5) instead of (-4, 5).
8. The equation of a circle is $(x+1)^2+(y-4)^2=64$. What is the circumference of the circle?
   Answer: $16\pi$ 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.
9. A circle has radius $6$. What is the area of the circle?
   Answer: $36\pi$ We need to find the area of a circle with radius 6. The area formula is A = πr². Substituting r = 6: A = π(6)² = 36π. Choice A incorrectly uses the circumference formula 2πr = 12π, while choice C uses 3πr instead of πr².
10. On a coordinate plane, a circle has equation $(x+1)^2+(y-4)^2=64$. What is the radius of the circle?
    Answer: $8$ 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.
11. What is the area of a circle with radius 12?
    Answer: $144\pi$ 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.
12. What is the measure of the arc with a central angle of 120° in a circle with radius 3?
    Answer: $2\pi$ 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.
13. What is the equation of a circle with center (0, 0) and radius 8?
    Answer: $x^2 + y^2 = 64$ We need the equation of a circle with center (0, 0) and radius 8. The standard form is (x - h)² + (y - k)² = r². With center at origin and r = 8: x² + y² = 8² = 64. Choice A incorrectly uses the radius instead of radius squared, while choices C and D include incorrect center shifts.
14. A circle has diameter $10$. What is the area of the circle?
    Answer: $25\pi$ We need to find the area of a circle with diameter 10. The area formula is A = πr², and since diameter = 10, radius = 5. Substituting r = 5: A = π(5)² = 25π. Choice A incorrectly uses the diameter in the area formula, while choice C uses the circumference formula.
15. Calculate the circumference of a circle with radius 15.
    Answer: $30\pi$ We need to find the circumference of a circle with radius 15. Using the circumference formula C = 2πr: C = 2π(15) = 30π. Choice A uses only the radius, choice C uses 3 times the radius, and choice D uses 4 times the radius.
16. A circle has a radius of 12. What is the area of the circle?
    Answer: $144\pi$ We need to find the area of a circle with radius 12. Using the area formula A = πr²: A = π(12)² = 144π. Choice B incorrectly uses the circumference formula 2πr, choice C uses an incorrect coefficient, and choice D uses radius times 3π.
17. A circle has area $64\pi$. What is the radius of the circle?
    Answer: $8$ We need to find the radius when the area is 64π. Using the area formula A = πr², we have 64π = πr². Dividing both sides by π gives 64 = r², so r = 8. Choice C (4) appears to be the result of taking the square root of 16 instead of 64.
18. A circle has diameter $12$. What is the area of the circle?
    Answer: $36\pi$ We need to find the area of a circle with diameter 12. The area formula is A = πr², and since diameter = 12, radius = 6. Substituting r = 6: A = π(6)² = 36π. Choice B incorrectly uses the circumference formula 2πr, while choice C uses the diameter instead of radius in the area formula.
19. A circle is given by the equation $(x-4)^2+(y+1)^2=64$. What is the radius of the circle?​​​
    Answer: $8$ 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.
20. Find the circumference of a circle with radius 14.
    Answer: $28\pi$ We need to find the circumference of a circle with radius 14. Using the circumference formula $C = 2\pi r$: $C = 2\pi(14) = 28\pi$. Choice A uses only the radius, choice C doubles the correct answer, and choice D uses 3 times the radius.

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