Circles
Help Questions
ACT Math › Circles
What is the area of a circle with diameter $10$ inches?
$10\pi$
$50\pi$
$100\pi$
$25\pi$
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.
What is the area of sector with central angle 120° in a circle with radius 8?
$\frac{64\pi}{6}$
$\frac{32\pi}{3}$
$\frac{64\pi}{3}$
$\frac{128\pi}{3}$
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.
On a coordinate plane, a circle has equation $(x+1)^2+(y-4)^2=64$. What is the radius of the circle?
$4$
$64$
$8$
$16$
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.
Find the area of a circle with radius 7.
$49\pi$
$14\pi$
$28\pi$
$21\pi$
Explanation
We need to find the area of a circle with radius 7. The area formula is $A = \pi r^2$. Substituting r = 7: $A = \pi(7)^2 = 49\pi$. Choice B incorrectly uses the circumference formula $2\pi r$, while choice C uses an incorrect coefficient.
What is the area of a circle with radius 12?
$144\pi$
$72\pi$
$36\pi$
$24\pi$
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.
If the circumference of a circle is $20\pi$, what is its radius?
5
10
15
20
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.
What is the radius of a circle if the circumference is $16\pi$?
4
8
16
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.
A circle has an area of $64\pi$. What is its diameter?
8
12
16
32
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.
A circle has radius $5$. What is the area of the circle?
$25\pi$
$50\pi$
$10\pi$
$5\pi$
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.
A circle has a diameter of 20. What is the area of the circle?
$400\pi$
$200\pi$
$100\pi$
$300\pi$
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.