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SAT · Math · Geometry and Trigonometry
Circles is part of Geometry and Trigonometry in the SAT Math section. The questions below are written in the style of the real test, each with the answer and a worked explanation.
Question types you will see here include center and radius from an equation, arc length from a central angle, sector area and central and inscribed angles.
Common traps: reading the signs of the center wrongly; reporting r² as the radius; not completing the square for both variables.
Question 1
The edge of the area covered by a weather radar is modeled in the xy-plane by the circle (x + 3)² + (y − 8)² = 49. What are the coordinates of the center of the circle?
(−3, 8)
(−3, −8)
(3, −8)
(3, 8)
Answer: A. (−3, 8)
The form (x − h)² + (y − k)² = r² has center (h, k). Here x + 3 = x − (−3) and the y-term is y − 8, so the center is (−3, 8) (A). D flips the sign of the x-coordinate, C flips both signs, and B flips only the y-coordinate.
Question 2
A bicycle wheel has a radius of 0.35 meter. Which of the following is closest to the number of turns the wheel makes as the bicycle rolls 22 meters along a straight road without slipping?
5
10
20
31
Answer: B. 10
Each turn moves the bicycle one circumference, 2π(0.35) = 0.7π ≈ 2.2 meters, so the wheel makes about 22 ÷ 2.2 = 10 turns (B). 20 divides by πr instead of 2πr, 5 uses the diameter 0.7 as the radius in 2πr, and 31 divides by the diameter 0.7 without π.
Question 3
A circle has a radius of 4. What is the area of a sector with a central angle of 135°?
6π
3π
12π
16π
Answer: A. 6π
Sector area = (θ/360)(πr²) = (135/360)(π·16) = 6π.
Question 4
The equation of a circle in the xy-plane is x² + y² - 2x + 8y + 8 = 0. What are the coordinates of the center of the circle?
(1, -4)
(-1, 4)
(-2, 8)
(-4, 1)
Answer: A. (1, -4)
Complete the square: (x - 1)² + (y + 4)² = 1 + 16 - (8) = 9. The center is (1, -4) and the radius is √9 = 3.
Question 5
In the xy-plane, the graph of x² + y² − 10x + c = 0, where c is a constant, is a circle that passes through the point (8, 6). What is the value of c?
Enter your own answer.
Answer: -20
Substituting x = 8 and y = 6 gives 64 + 36 − 80 + c = 0, so 20 + c = 0 and c = −20. The equation is then (x − 5)² + y² = 45, a real circle. Dropping the sign of the −10x term gives 64 + 36 + 80 + c = 0 and c = −180.
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