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ACT Math · Practice Questions
25 ACT Circle Questions
You Must Master Before Test Day
These 25 questions cover every circle topic the ACT tests — circumference and area, arc length and sector area, chords and tangents, inscribed and central angles, and the equation of a circle. Work through them all, check your answers, and read the explanations.
Circumference & Area
Arc Length & Sectors
Chords & Tangents
Inscribed & Central Angles
Equation of a Circle
Applied Problems
Category 1
Circumference & Area
Questions 1–5
1
Circumference
A circle has a radius of 7 cm. What is its circumference in terms of π?
A 7π cm
B 14π cm
C 49π cm
D 21π cm
Solution
C = 2πr = 2π(7) = 14π cm
⚠️ Circumference = 2πr (or πd). Using diameter instead of radius without adjusting the formula is the most common error here.
2
Area
A circle has a diameter of 18 inches. What is its area in terms of π?
A 18π in²
B 81π in²
C 324π in²
D 9π in²
Solution
Radius = diameter / 2 = 18 / 2 = 9
A = πr² = π(9)² = 81π in²
⚠️ Always convert diameter to radius first. Plugging the diameter directly into πr² gives an answer 4 times too large — a common ACT trap answer.
3
Circumference → Radius
The circumference of a circle is 30π meters. What is the radius?
A 10 m
B 15 m
C 30 m
D 60 m
Solution
C = 2πr
30π = 2πr
r = 30π / 2π = 15 m
💡 When π appears on both sides of an equation, it cancels out — you only need to work with the coefficients.
4
Area → Radius
A circle has an area of 64π square feet. What is its diameter?
A 8 ft
B 16 ft
C 32 ft
D 64 ft
Solution
A = πr²
64π = πr²
r² = 64 → r = 8 ft
Diameter = 2r = 16 ft
⚠️ The question asks for the diameter, not the radius. Solving for r and stopping there is a common mistake — always re-check what's actually being asked.
5
Ratio of Areas
Circle A has radius 3 and Circle B has radius 6. What is the ratio of the area of Circle A to the area of Circle B?
A 1:2
B 1:4
C 1:3
D 1:6
Solution
Area A = π(3)² = 9π
Area B = π(6)² = 36π
Ratio = 9π : 36π = 1 : 4
💡 When radius doubles, area increases by a factor of 4 (2²), not 2. Area scales with the SQUARE of the radius ratio.
Category 2
Arc Length & Sector Area
Questions 6–10
6
Arc Length
A circle has a radius of 12 cm. What is the length of an arc with a central angle of 60°?
A 2π cm
B 4π cm
C 6π cm
D 8π cm
Solution
Arc length = (θ/360) × 2πr
= (60/360) × 2π(12)
= (1/6) × 24π = 4π cm
⚠️ Arc length is a FRACTION of the full circumference. The fraction equals (central angle)/360°.
7
Sector Area
A circle has a radius of 9 inches. What is the area of a sector with a central angle of 80°?
A 16.2π in²
B 18π in²
C 9π in²
D 20.25π in²
Solution
Sector area = (θ/360) × πr²
= (80/360) × π(9)²
= (80/360)(81π) = 16.2π in²
💡 Sector area is a FRACTION of the full circle's area, using the same fraction as arc length: (central angle)/360°.
8
Arc Length → Angle
A circle has a radius of 10 cm. An arc has a length of 5π cm. What is the measure of the central angle?
A 45°
B 60°
C 90°
D 120°
Solution
Circumference = 2π(10) = 20π
Fraction of circle = 5π / 20π = 1/4
Angle = (1/4) × 360° = 90°
💡 Find what fraction the arc is of the full circumference first, then apply that same fraction to 360°.
9
Sector Area → Radius
A sector with a central angle of 90° has an area of 16π square units. What is the radius of the circle?
A 4
B 6
C 8
D 16
Solution
(90/360) × πr² = 16π
(1/4)r² = 16
r² = 64 → r = 8
⚠️ Set up the sector area equation first, then solve for r² before taking the square root — don't try to isolate r too early.
10
Arc Length (Word Problem)
A clock has a minute hand 8 cm long. How far does the tip of the minute hand travel in 20 minutes?
A (8/3)π cm
B (16/3)π cm
C 8π cm
D 16π cm
Solution
20 minutes = 20/60 = 1/3 of a full rotation (360°)
Circumference = 2π(8) = 16π
Arc length = (1/3)(16π) = (16/3)π cm
💡 Convert the time fraction to a fraction of the full circle (60 minutes = 360°), then apply it to the circumference.
Category 3
Chords & Tangents
Questions 11–15
11
Chord & Perpendicular Bisector
A chord of a circle is 16 units long, and the radius is 10 units. What is the distance from the center to the chord?
A 4
B 6
C 8
D 12
Solution
A radius perpendicular to a chord bisects it: half-chord = 16/2 = 8
Right triangle: radius² = (half-chord)² + distance²
10² = 8² + d² → 100 = 64 + d² → d²=36 → d=6
⚠️ The perpendicular from the center to a chord always bisects the chord. This creates a right triangle with the radius as hypotenuse.
12
Tangent Line
A tangent line touches a circle at point P. If the radius to point P is drawn, what is the angle between the radius and the tangent line?
A 45°
B 60°
C 90°
D It depends on the circle's size
Solution
A tangent line is always perpendicular to the radius drawn to the point of tangency.
Angle = 90°, regardless of the circle's size.
💡 This radius-tangent perpendicularity rule is one of the most frequently used circle facts on the ACT — memorize it cold.
13
Tangent Length
A point P is 13 units from the center of a circle with radius 5. What is the length of the tangent segment from P to the circle?
A 8
B 10
C 12
D 18
Solution
The radius to the tangent point is perpendicular to the tangent line, forming a right triangle.
tangent² + radius² = distance²
t² + 5² = 13² → t² = 169 − 25 = 144 → t = 12
⚠️ This is a 5-12-13 Pythagorean triple in disguise — recognizing common triples saves calculation time.
14
Two Tangents from a Point
From an external point, two tangent segments are drawn to a circle. If one tangent segment has length 11, what is the length of the other?
A Cannot be determined
B 5.5
C 11
D 22
Solution
Two tangent segments drawn from the same external point to a circle are always congruent.
Other tangent length = 11
💡 This is a standalone circle theorem: tangents from a common external point are always equal in length, regardless of the circle's size.
15
Chord Length
A circle has radius 9. A chord is drawn 6 units from the center. What is the length of the chord?
A 2√45
B 6√5
C 12
D 15
Solution
half-chord² + 6² = 9²
half-chord² = 81 − 36 = 45
half-chord = √45 = 3√5
Full chord = 2 × 3√5 = 6√5
⚠️ Don't forget to DOUBLE the half-chord at the end — the Pythagorean step only gives you half the chord length.
Category 4
Inscribed & Central Angles
Questions 16–19
16
Inscribed Angle Theorem
A central angle measures 80°. What is the measure of an inscribed angle that intercepts the same arc?
A 20°
B 40°
C 80°
D 160°
Solution
Inscribed angle = (1/2) × central angle
= (1/2)(80°) = 40°
⚠️ Inscribed Angle Theorem: an inscribed angle is always HALF the central angle that subtends the same arc.
17
Angle in a Semicircle
Triangle ABC is inscribed in a circle with AC as a diameter. What is the measure of angle B?
A 45°
B 60°
C 90°
D Cannot be determined
Solution
An angle inscribed in a semicircle (where the opposite side is a diameter) is always 90°.
This follows from the Inscribed Angle Theorem: the central angle for a diameter is 180°, so the inscribed angle is 90°.
💡 This special case — "angle inscribed in a semicircle is a right angle" — appears constantly on the ACT and is worth memorizing as its own fact.
18
Central Angle & Arc Measure
A central angle intercepts an arc. If the arc measures 130°, what is the measure of the central angle?
A 65°
B 130°
C 230°
D 260°
Solution
The measure of a central angle always EQUALS the measure of its intercepted arc.
Central angle = 130°
⚠️ Don't confuse this with the inscribed angle rule. Central angle = arc measure (1:1). Inscribed angle = half the arc measure (1:2).
19
Inscribed Quadrilateral
A quadrilateral is inscribed in a circle. One angle measures 75°. What is the measure of the opposite angle?
A 75°
B 95°
C 105°
D 285°
Solution
Opposite angles of an inscribed (cyclic) quadrilateral are supplementary — they sum to 180°.
Opposite angle = 180° − 75° = 105°
💡 Cyclic quadrilateral rule: opposite angles always sum to 180°. This is a less common but high-value fact on harder ACT geometry questions.
Category 5
Equation of a Circle
Questions 20–22
20
Equation of a Circle
What is the equation of a circle with center (3, −2) and radius 5?
A (x−3)² + (y+2)² = 5
B (x+3)² + (y−2)² = 25
C (x−3)² + (y+2)² = 25
D (x−3)² − (y+2)² = 25
Solution
Standard form: (x−h)² + (y−k)² = r²
Center (h,k) = (3, −2): use (x−3) and (y−(−2)) = (y+2)
Radius 5: r² = 25
(x−3)² + (y+2)² = 25
⚠️ Watch the sign flip: a center of (3,−2) produces (y+2) in the equation, not (y−2). Also remember r² goes on the right, not r itself.
21
Center & Radius from Equation
The equation (x+4)² + (y−1)² = 36 represents a circle. What are its center and radius?
A Center (4,−1), radius 6
B Center (−4,1), radius 6
C Center (−4,1), radius 36
D Center (4,−1), radius 18
Solution
(x+4)² = (x−(−4))² → h = −4
(y−1)² → k = 1
r² = 36 → r = 6
Center: (−4, 1), Radius: 6
💡 Always take the square root of the right-hand side to find the radius — don't use 36 directly as the radius.
22
Completing the Square
What is the radius of the circle given by x² + y² − 6x + 4y − 12 = 0?
A 3
B 4
C 5
D 25
Solution
Group: (x²−6x) + (y²+4y) = 12
Complete the square: (x−3)²−9 + (y+2)²−4 = 12
(x−3)² + (y+2)² = 25
r²=25 → r=5
⚠️ When completing the square, take half the x-coefficient and half the y-coefficient, square each, and add both to BOTH sides of the equation.
Category 6
Applied & Multi-Step Problems
Questions 23–25
23
Inscribed Square
A square is inscribed in a circle of radius 6. What is the area of the square?
A 36
B 72
C 144
D 108
Solution
The diagonal of the square = diameter of circle = 12
For a square: diagonal = side√2 → side = 12/√2 = 6√2
Area = side² = (6√2)² = 72
💡 An inscribed square's diagonal always equals the circle's diameter. This connects circle geometry to the 45-45-90 triangle relationship.
24
Shaded Region
A square with side length 10 has a circle inscribed inside it (the circle touches all four sides). What is the area of the region inside the square but outside the circle, in terms of π?
A 100 − 25π
B 100 − 100π
C 25π − 100
D 100 − 50π
Solution
Circle's diameter = side of square = 10 → radius = 5
Square area = 10² = 100
Circle area = π(5)² = 25π
Shaded area = 100 − 25π
⚠️ "Inscribed inside, touching all sides" means the circle's diameter equals the square's side length — not the diagonal.
25
Distance & Circle Equation
A circle has equation (x−2)² + (y−5)² = 50. Does the point (9, 10) lie on, inside, or outside the circle?
A On the circle
B Inside the circle
C Outside the circle
D Cannot be determined
Solution
Center (2,5), r²=50.
Distance² from center to (9,10): (9−2)²+(10−5)² = 49+25 = 74
Since 74 > 50, the point is OUTSIDE the circle.
⚠️ Compare the computed distance² to r²: equal means on the circle, less means inside, greater means outside. No need to take square roots.
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