SAT 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 SAT 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. 8
2. $16\pi$
3. $2\pi$
4. $8\pi$ (correct answer)

Explanation: Sector area with the central angle in radians is (1/2)($r^2$)(theta). With r = 8 and theta = pi/4, that gives (1/2)(64)(pi/4) = 32(pi/4) = 8pi. A sanity check agrees: the full circle has area 64pi, and pi/4 is one-eighth of the full 2pi turn, so the sector is 64pi divided by 8, or 8pi. The 16pi value drops the factor of 1/2, computing 64 times pi/4 and doubling the true sector. The 2pi value is the arc length rather than the area, since r times theta equals 8 times pi/4, a length measure instead of a square measure. The bare 8 carries no pi at all and simply matches the radius, a sign the area formula was abandoned partway through.

Question 2

1. $50$
2. $100$
3. $300$
4. $150$ (correct answer)

Explanation: This question asks how far a runner travels on a circular track. The runner completes 3/8 of a lap on a track with circumference 400 meters. Distance traveled = (3/8) × 400 = 3 × 50 = 150 meters. A common error is misunderstanding what fraction of a lap means - it's the fraction of the total circumference. When dealing with partial laps, multiply the fraction by the total circumference to find the distance.

Question 3

1. Center $(-3,4)$, $r=5$
2. Center $(3,-4)$, $r=5$ (correct answer)
3. Center $(-3,-4)$, $r=25$
4. Center $(3,4)$, $r=25$

Explanation: This question asks to identify the center and radius from a circle's standard form equation. The standard form is (x-h)² + (y-k)² = r², where (h,k) is the center and r is the radius. From (x-3)² + (y+4)² = 25, we have h = 3, k = -4 (note: y+4 = y-(-4)), and r² = 25, so r = 5. A common error is confusing the signs: the center is (3,-4), not (-3,4). Remember that (x-h) means the x-coordinate is h, and (y-k) means the y-coordinate is k.

Question 4

1. $(2,5)$ (correct answer)
2. $(2,8)$
3. $(5,2)$
4. (-1,5)

Explanation: This question asks which point lies on the given circle. The circle has equation (x+1)² + (y-5)² = 9, with center (-1, 5) and radius 3. To check if a point lies on the circle, substitute its coordinates and verify the equation holds. For point (2, 5): (2+1)² + (5-5)² = 3² + 0² = 9 ✓. This confirms (2, 5) lies on the circle. A quick strategy is to check points that are horizontally or vertically aligned with the center, as these calculations are simpler.

Question 5

1. $324\pi$
2. $81\pi$ (correct answer)
3. $36\pi$
4. $162\pi$

Explanation: This question asks for the area of a circle given its diameter. The area formula is A = πr², where r is the radius. Since diameter = 18 cm, the radius = 18/2 = 9 cm. Substituting: A = π(9)² = π(81) = 81π square centimeters. A common mistake is using the diameter directly in the formula instead of converting to radius first. Always remember to divide the diameter by 2 to get the radius before calculating area.

Question 6

1. $5$ (correct answer)
2. $10$
3. $\sqrt{25}$
4. $\sqrt{5}$

Explanation: This question requires finding the radius from a circle equation in general form. First, complete the square to convert to standard form. Starting with x² + y² + 6x - 8y = 0, group x and y terms: (x² + 6x) + (y² - 8y) = 0. Complete the square: (x² + 6x + 9) + (y² - 8y + 16) = 9 + 16, which gives (x+3)² + (y-4)² = 25. Therefore, r² = 25, so r = 5. A common mistake is forgetting to add the completing-the-square terms to both sides. When completing the square, always balance the equation.

Question 7

1. $48\pi$
2. 12
3. $12\pi$ (correct answer)
4. 2160

Explanation: Sector area is $\frac{120}{360}\cdot\pi\cdot 6^2 = 12\pi$. Choice A uses the diameter in place of the radius, B omits $\pi$, and D treats 120 as radians.

Question 8

1. $13$ (correct answer)
2. $\sqrt{244}$
3. $17$
4. $22$

Explanation: This problem involves finding the radius when given a chord length and the perpendicular distance from center to chord. When a perpendicular from the center meets a chord, it bisects the chord, creating a right triangle with the radius as hypotenuse. Half the chord length is 5 units, and the perpendicular distance is 12 units. Using the Pythagorean theorem: $r^2 = 5^2 + 12^2 = 25 + 144 = 169$, so $r = sqrt{169} = 13$ units. The key insight is recognizing the right triangle formed and that the perpendicular bisects the chord.

Question 9

1. $1200\pi$ m
2. $300\pi$ m
3. $200\pi$ m
4. $600\pi$ m (correct answer)

Explanation: This question asks for the total distance traveled by a runner completing 3 laps on a circular track with diameter 200 m. First, find the circumference using $C = \pi d$, where $d$ is the diameter: $C = \pi(200) = 200\pi$ m. For 3 complete laps, multiply by 3: Total distance = $3 \times 200\pi = 600\pi$ m. A common error is using the radius formula $C = 2\pi r$ without converting diameter to radius, or forgetting to multiply by the number of laps. When given diameter, you can use $C = \pi d$ directly instead of converting to radius first.

Question 10

1. $55^\circ$ (correct answer)
2. $110^\circ$
3. $220^\circ$
4. $35^\circ$

Explanation: This question asks for the measure of inscribed angle $\angle ACB$ that intercepts arc AB with measure $110^\circ$. The inscribed angle theorem states that an inscribed angle is half the measure of its intercepted arc. Therefore, $\angle ACB = \frac{1}{2} \times 110^\circ = 55^\circ$. A common error is confusing inscribed angles with central angles, which would give $110^\circ$. Remember: inscribed angles are always half the measure of their intercepted arcs, while central angles equal their intercepted arcs.

Question 11

1. $14\pi$
2. $28\pi$ (correct answer)
3. $49\pi$
4. $392\pi$

Explanation: This question requires finding circumference given the area. From A = πr² = 196π, we get r² = 196, so r = 14 feet. The circumference is C = 2πr = 2π(14) = 28π feet. A common mistake is taking the square root of 196π instead of just 196. Remember to isolate r² by dividing out π first, then take the square root to find r.

Question 12

1. $10$ (correct answer)
2. $8$
3. $14$
4. $12$

Explanation: This question involves a chord and its perpendicular distance from the center. When a perpendicular from the center bisects a chord, it creates a right triangle. The chord AB = 16 units is bisected into two 8-unit segments, and the perpendicular distance is 6 units. Using the Pythagorean theorem: r² = 8² + 6² = 64 + 36 = 100, so r = 10. A key insight is that the perpendicular from the center always bisects the chord. Draw the diagram to visualize the right triangle formed.

Question 13

1. $x^2+y^2=52$
2. $(x+2)^2+(y-5)^2=52$
3. $(x-2)^2+(y+5)^2=52$
4. $(x-2)^2+(y-5)^2=52$ (correct answer)

Explanation: To find the equation of a circle with center (2,5) passing through (8,1), we first need the radius. The radius is the distance from center to the given point: r = √[(8-2)² + (1-5)²] = √[6² + (-4)²] = √[36 + 16] = √52. The standard form equation is (x-h)² + (y-k)² = r², so with center (2,5) and r² = 52, the equation is (x-2)² + (y-5)² = 52. Remember to keep r² = 52 rather than trying to simplify √52, as the equation uses r².

Question 14

1. $8$
2. $9$ (correct answer)
3. $10$
4. $12$

Explanation: This problem involves the intersecting chords theorem, which states that when two chords intersect inside a circle, the products of their segments are equal: AE × EB = CE × ED. Given AE = 3, EB = 12, and CE = 4, we can solve for ED: 3 × 12 = 4 × ED, so 36 = 4 × ED, giving ED = 9. This theorem is a powerful tool for finding unknown segments when chords intersect. Always identify which segments belong to which chord before applying the formula.

Question 15

1. $45^\circ$
2. $60^\circ$
3. $90^\circ$ (correct answer)
4. $180^\circ$

Explanation: The question asks for the central angle of a sector with radius 5 in and area 25π/4 in². The sector area formula is (θ/360) × πr²; solve for θ. With r = 5, full area π × 25 = 25π, so (θ/360) × 25π = 25π/4, θ/360 = 1/4, θ = 90°. The fraction simplifies directly. A common error is using arc length or forgetting to multiply by 360, leading to 45° or 180°. Another is confusing r with diameter, doubling area. Emphasize isolating θ correctly and using radius in r². Rearrange the formula before substituting to avoid errors.

Question 16

1. $(x-2)^2+(y+1)^2=20$
2. $(x+2)^2+(y-1)^2=52$
3. $(x-2)^2+(y+1)^2=52$ (correct answer)
4. $(x+2)^2+(y-1)^2=20$

Explanation: The question asks for the equation of a circle with center (2, -1) passing through (8, 3). The standard form is (x-h)² + (y-k)² = r², where (h,k) is the center; find r using the distance from center to the point. Calculate r = √[(8-2)² + (3 - (-1))²] = √[6² + 4²] = √[36 + 16] = √52. Thus, the equation is (x-2)² + (y+1)² = 52. A common error is switching signs for the center, like using (y-1) instead of (y+1), or squaring incorrectly to get 20 instead of 52. Emphasize radius as the distance, not diameter. Plug the point back into the equation to verify it satisfies it.

Question 17

1. $11$
2. $13$ (correct answer)
3. $12$
4. $17$

Explanation: The question involves finding the radius of a circle given a chord length and the perpendicular distance from the center to the chord. When a perpendicular from the center meets a chord, it bisects the chord, creating a right triangle with the radius as hypotenuse. Half the chord length is $\frac{24}{2} = 12$, and the perpendicular distance is 5. Using the Pythagorean theorem: $r^2 = 12^2 + 5^2 = 144 + 25 = 169$, so $r = 13$. A common error is using the full chord length instead of half in the Pythagorean theorem. Remember that the perpendicular from the center always bisects the chord.

Question 18

1. $10$
2. $15$ (correct answer)
3. $20$
4. $30$

Explanation: The question asks for the radius of a circle given a central angle and arc length. The formula relating arc length, radius, and central angle (in radians) is $s = r\theta$, where $s$ is arc length, $r$ is radius, and $\theta$ is the angle in radians. First, convert $120°$ to radians: $120° = 120 \cdot \frac{\pi}{180} = \frac{2\pi}{3}$ radians. Now substitute into the formula: $10\pi = r \cdot \frac{2\pi}{3}$, so $r = \frac{10\pi}{\frac{2\pi}{3}} = 10\pi \cdot \frac{3}{2\pi} = 15$ inches. A common error is forgetting to convert degrees to radians before using the arc length formula. Always check whether angles are given in degrees or radians.

Question 19

1. $9$
2. $8$
3. $6$ (correct answer)
4. $2$

Explanation: A chord of length 16 in a circle with radius 10 is perpendicular to a radius at its midpoint. This creates a right triangle where the hypotenuse is the radius (10), one leg is half the chord (8), and the other leg is the distance from center to chord. Using the Pythagorean theorem: $10^2 = 8^2 + d^2$, so $100 = 64 + d^2$, giving $d^2 = 36$ and $d = 6$. The key insight is that when a radius is perpendicular to a chord, it bisects that chord.

Question 20

1. $625\pi$ (correct answer)
2. $250\pi$
3. $100\pi$
4. $25\pi$

Explanation: Given a circle with circumference $50\pi$ inches, we need to find its area. From $C = 2\pi r = 50\pi$, we can solve for radius: $r = \frac{50\pi}{2\pi} = 25$ inches. The area formula is $A = \pi r^2 = \pi(25)^2 = 625\pi$ square inches. A common mistake is confusing radius and diameter; here the radius is 25, not the diameter. When working backwards from circumference, always solve for radius first before calculating area.