Circles Flashcards

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

Answer: 7

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

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

Answer: (3, −5)

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

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

Answer: 10π

Circumference = 2πr = 2π(5) = 10π.

A circle has radius 6. What is its area?

Answer: 36π

Area = πr² = π(6)² = 36π.

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

Answer: 5π

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

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

Answer: 27π

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

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

Answer: 70°

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

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

Answer: (x−5)² + y² = 25

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

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

Answer: 47°

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

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

Answer: C²/(4π)

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

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

Answer: 25

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

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?

Answer: 75π

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

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

Answer: 6

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

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

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.

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

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.

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?

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.

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

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.

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?

Answer: 5π

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

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

Answer: 5π

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

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

Answer: 80°

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