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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
x² + y² − 6x + 4y − 12 = 0 The equation above defines a circle in the xy-plane. What is the radius of the circle?
Enter your own answer.
Answer: 5
Complete the square: (x² − 6x + 9) + (y² + 4y + 4) = 12 + 9 + 4, so (x − 3)² + (y + 2)² = 25 and the radius is √25 = 5. Using 12 as r² (forgetting to add 9 and 4) gives about 3.46.
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 circular turntable of a mountain cable-car station has a radius of 12 metres. Two neighbouring cabins are parked so that the arc of the turntable's edge between them has length 4π metres. What is the area, in square metres, of the sector of the turntable between the two cabins and its centre?
24π
48π
72π
144π
Answer: A. 24π
The full circumference is 24π, so the arc is 1/6 of the circle and the sector area is (1/6)(144π) = 24π. B multiplies radius by arc length without the ½ (12 × 4π). C treats the sector as a half circle. D is the whole turntable's area.
Question 5
A circle in the xy-plane has its centre at (3, −1) and passes through the point (7, 2). Which of the following is an equation of the circle?
(x − 3)² + (y + 1)² = 25
(x + 3)² + (y − 1)² = 25
(x − 3)² + (y + 1)² = 5
(x − 3)² + (y + 1)² = 7
Answer: A. (x − 3)² + (y + 1)² = 25
The radius is the distance from (3, −1) to (7, 2): √(4² + 3²) = 5, so r² = 25 (A). Choice B has the wrong centre signs; C writes r instead of r²; D adds the differences 4 + 3 instead of squaring them.
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