Bluebook SAT Test Circles — Questions and Answers

Question 1: What is the radius of the circle with equation x² + y² = 49?

7 (Correct answer)
49
√49/2
24.5

Correct answer: 7

The standard form x² + y² = r² gives r² = 49, so r = 7.

Question 2: The center of the circle (x − 3)² + (y + 5)² = 16 is:

(3, −5) (Correct answer)
(−3, 5)
(3, 5)
(−3, −5)

Correct answer: (3, −5)

In (x − h)² + (y − k)² = r², the center is (h, k) = (3, −5).

Question 3: A circle has center (0, 0) and radius 5. What is the circumference?

10π (Correct answer)
25π
5π
2π

Correct answer: 10π

Circumference = 2πr = 2π(5) = 10π.

Question 4: A circle has radius 6. What is its area?

36π (Correct answer)
12π
6π
72π

Correct answer: 36π

Area = πr² = π(6)² = 36π.

Question 5: An arc of a circle with radius 10 subtends a central angle of 90°. What is the arc length?

5π (Correct answer)
10π
25π
2.5π

Correct answer: 5π

Arc length = (θ/360°)·2πr = (90/360)·2π(10) = (1/4)(20π) = 5π.

Question 6: A central angle of a circle measures 120° and the radius is 9. What is the area of the corresponding sector?

27π (Correct answer)
54π
18π
81π

Correct answer: 27π

Sector area = (θ/360°)·πr² = (120/360)·π(81) = (1/3)(81π) = 27π.

Question 7: An inscribed angle in a circle measures 35°. What is the central angle that subtends the same arc?

70° (Correct answer)
35°
17.5°
140°

Correct answer: 70°

The Inscribed Angle Theorem states the central angle is twice the inscribed angle: 2 × 35° = 70°.

Question 8: A circle passes through the origin. Its center is at (5, 0). What is the equation of the circle?

x² + y² = 25
(x−5)² + y² = 25 (Correct answer)
x² + y² = 5
(x+5)² + y² = 25

Correct answer: (x−5)² + y² = 25

The radius = distance from (5,0) to (0,0) = 5. Equation: (x−5)² + y² = 25.

Question 9: In a circle, an inscribed angle measures 47°. Another inscribed angle intercepts the same arc. What is its measure?

94°
47° (Correct answer)
133°
23.5°

Correct answer: 47°

Inscribed angles that intercept the same arc are equal in measure.

Question 10: A circle has circumference C. What is the area of the circle in terms of C?

C²/π
C²/(2π)
C²/(4π) (Correct answer)
πC²

Correct answer: C²/(4π)

C = 2πr so r = C/(2π). Area = πr² = π(C/(2π))² = πC²/(4π²) = C²/(4π).

Question 11: The circle (x−2)² + (y+3)² = r² passes through the point (−1, 1). What is r²?

9
13
25 (Correct answer)
18

Correct answer: 25

r² = (−1−2)² + (1−(−3))² = (−3)² + 4² = 9 + 16 = 25.

Question 12: Two circles have radii in the ratio 2:5. The area of the smaller circle is 12π. What is the area of the larger circle?

30π
75π (Correct answer)
50π
60π

Correct answer: 75π

Area ratio = (2:5)² = 4:25. If smaller area = 12π, larger = 12π × (25/4) = 75π.

Question 13: A sector has a central angle of 3π/4 radians and arc length 9π/2. What is the radius?

4
6 (Correct answer)
3
12

Correct answer: 6

Arc length = rθ, so 9π/2 = r × (3π/4). r = (9π/2) ÷ (3π/4) = (9π/2) × (4/(3π)) = 6.

Question 14: A circle has center O and radius 10. Chord AB has length 16. What is the distance from O to the midpoint of AB?

6
8
√(100−64)
Both A and C (Correct answer)

Correct answer: Both A and C

The perpendicular from O to chord AB meets AB at its midpoint, and the distance = √(10² − 8²) = √(100−64) = √36 = 6. Both answer A (6) and answer C (√36 = 6) are correct.

Question 15: The equation x² + y² − 6x + 2y − 6 = 0 represents a circle. What is the radius?

4
√6
√16
Both A and C (Correct answer)

Correct answer: Both A and C

Completing the square: (x−3)² − 9 + (y+1)² − 1 − 6 = 0 → (x−3)² + (y+1)² = 16. Radius = 4 = √16. Both A and C are equivalent.

Question 16: Point P is outside a circle of radius 7. A tangent from P touches the circle at T and the distance PT = 24. What is the distance from P to the center O?

17
25
√(576+49)
Both B and C (Correct answer)

Correct answer: Both B and C

OT ⊥ PT (radius to tangent point), so PO² = PT² + OT² = 576 + 49 = 625. PO = 25 = √625. Both B and C are equivalent.

Question 17: A circle has center (0, 0) and radius r. A horizontal line y = 3 is tangent to the circle. What is r?

3 (Correct answer)
9
√3
6

Correct answer: 3

A line is tangent to a circle when the distance from the center to the line equals the radius. Distance from (0,0) to y=3 is 3, so r = 3.

Question 18: What is the area of the region inside circle (x−1)² + (y−1)² = 9 but outside circle (x−1)² + (y−1)² = 4?

5π (Correct answer)
13π
5
π√5

Correct answer: 5π

Radii are √9 = 3 and √4 = 2. Area of annulus = π(3² − 2²) = π(9−4) = 5π.

Question 19: A central angle of 72° subtends an arc. If the radius of the circle is 5, what is the area of the sector?

2π
5π (Correct answer)
10π
π

Correct answer: 5π

Sector area = (θ/360°)·πr² = (72/360)·π(25) = (1/5)(25π) = 5π.

Question 20: An inscribed angle in a circle intercepts an arc of 160°. What is the measure of the inscribed angle?

160°
320°
80° (Correct answer)
40°

Correct answer: 80°

An inscribed angle equals half the intercepted arc: 160°/2 = 80°.

Question 21: The circle x² + y² = 100 intersects the positive x-axis and positive y-axis at points A and B respectively. What is the length of chord AB?

10
10√2 (Correct answer)
20
100

Correct answer: 10√2

A = (10, 0) and B = (0, 10). AB = √((10−0)² + (0−10)²) = √(100+100) = 10√2.

Question 22: A circle has equation (x + 3)² + (y − 5)² = 49. Which transformation maps this circle to the unit circle x² + y² = 1?

Translate (3,−5), then dilate by factor 1/7 (Correct answer)
Translate (−3, 5), then dilate by factor 1/7
Dilate by 7, then translate (3, −5)
Translate (3, −5), then dilate by factor 7

Correct answer: Translate (3,−5), then dilate by factor 1/7

First shift center (−3, 5) to origin by translating (+3, −5), giving x² + y² = 49, then shrink radius 7 to 1 by scaling by 1/7.

Question 23: A wheel of radius 2 feet rolls along a flat road without slipping. How far has the center of the wheel traveled after the wheel makes 5 complete rotations?

10π feet
20π feet (Correct answer)
5π feet
40π feet

Correct answer: 20π feet

Each rotation the wheel travels one circumference = 2π(2) = 4π feet. Five rotations = 5 × 4π = 20π feet.

Question 24: In a circle with radius 13, a chord is drawn 5 units from the center. What is the length of the chord?

12
24
√(169−25)
Both B and C (Correct answer)

Correct answer: Both B and C

Half-chord = √(13²−5²) = √(169−25) = √144 = 12. Full chord = 24. Both B and C are equal.

Question 25: Circle O has center (0, 0) and radius 5. Line y = 3 intersects the circle. What are the x-coordinates of the intersection points?

±4 (Correct answer)
±3
±√34
±√(25−9) = ±4

Correct answer: ±4

Substituting y = 3: x² + 9 = 25, so x² = 16 and x = ±4.

Question 26: A sector has radius 6 and central angle 150°. What is the area of the sector?

15π (Correct answer)
36π
12π
6π

Correct answer: 15π

Area = (150/360) × π(6)² = (5/12) × 36π = 15π.

Question 27: The diameter of a circle is the segment from (−4, 1) to (6, 1). What is the equation of the circle?

(x−1)² + (y−1)² = 25 (Correct answer)
(x−1)² + (y−1)² = 100
(x+4)² + (y+1)² = 25
x² + (y−1)² = 25

Correct answer: (x−1)² + (y−1)² = 25

Center = midpoint = ((−4+6)/2, (1+1)/2) = (1, 1). Radius = distance from center to endpoint = √((6−1)²+0²) = 5. Equation: (x−1)² + (y−1)² = 25.

Question 28: In a circle with center O and radius r, two radii OA and OB form a central angle of 90°. What is the length of chord AB in terms of r?

r
r√2 (Correct answer)
r√3
2r

Correct answer: r√2

With OA = OB = r and angle AOB = 90°, triangle OAB is an isosceles right triangle. AB = r√2.

Bluebook SAT Test Circles

What is the radius of the circle with equation x² + y² = 49?

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