Circles show up in every geometry course and on both the SAT and ACT. The good news is that almost every circle problem comes from a handful of formulas and one idea: an arc or sector is just a fraction of the whole circle.
Vocabulary
- Radius (r): center to edge.
- Diameter (d): edge to edge through the center. d = 2r.
- Chord: any segment with both ends on the circle. The diameter is the longest chord.
- Tangent: a line touching the circle at exactly one point. It is perpendicular to the radius at that point.
- Arc: a piece of the circle’s edge.
- Sector: a “pizza slice” bounded by two radii and an arc.
- Central angle: an angle with its vertex at the center.
Circumference and area
| Measure | Formula |
|---|---|
| Circumference | C = 2πr = πd |
| Area | A = πr² |
Example A circle has diameter 10 cm. Find its circumference and area.
- r = 5 cm.
- C = 2π(5) = 10π ≈ 31.42 cm.
- A = π(5²) = 25π ≈ 78.54 cm².
Use the radius in the area formula
With a diameter of 10, the area is π × 5², not π × 10². Using the diameter makes the answer four times too big.
Going backwards is common too. If the area is 49π, then r² = 49 and r = 7. If the circumference is 18π, then 2r = 18 and r = 9.
Arc length and sector area in degrees
An arc with central angle θ is θ/360 of the whole circle. So is its sector.
- Arc length = (θ/360) × 2πr
- Sector area = (θ/360) × πr²
Example A circle has radius 12. Find the length of an arc with central angle 60°, and the area of its sector.
- Fraction of the circle: 60/360 = 1/6.
- Arc length: (1/6)(2π × 12) = 4π ≈ 12.57.
- Sector area: (1/6)(π × 144) = 24π ≈ 75.40.
Arc length and sector area in radians
In radians the formulas are simpler, which is a big reason radians exist. With θ in radians:
- Arc length: s = rθ
- Sector area: A = ½r²θ
Example A sector has radius 6 and central angle 2π/3. Find its arc length and area.
- s = 6 × 2π/3 = 4π.
- A = ½ × 36 × 2π/3 = 12π.
The unit circle guide covers converting between degrees and radians.
Angles in circles
- A central angle equals the arc it cuts off. A 70° central angle intercepts a 70° arc.
- An inscribed angle has its vertex on the circle. It equals half the arc it intercepts. An inscribed angle on a 70° arc is 35°.
- An inscribed angle that intercepts a semicircle (a diameter) is always 90°.
- Two inscribed angles that intercept the same arc are equal.
- A tangent meets the radius at 90°. That right angle often sets up a Pythagorean theorem problem.
Example A tangent from point P touches a circle of radius 5 at T. The distance from P to the center O is 13. How long is PT?
- OT is a radius and is perpendicular to the tangent, so triangle OTP has a right angle at T.
- PT² + 5² = 13², so PT² = 169 − 25 = 144.
- PT = 12.
The equation of a circle
A circle with center (h, k) and radius r is every point at distance r from the center. Using the distance formula gives:
(x − h)² + (y − k)² = r²
Example Write the equation of the circle with center (3, −2) and radius 4
(x − 3)² + (y + 2)² = 16.
Note the sign flips: y − (−2) becomes y + 2, and the right side is r², not r.
Example Find the center and radius of (x + 1)² + (y − 5)² = 20
Center (−1, 5). Radius √20 = 2√5 ≈ 4.47. The radicals guide shows how to simplify √20.
Completing the square
Sometimes the equation is expanded, and you need to complete the square in x and in y to read the center and radius.
Example Find the center and radius of x² + y² − 6x + 4y − 12 = 0
- Group: (x² − 6x) + (y² + 4y) = 12.
- Half of −6 is −3, and (−3)² = 9. Half of 4 is 2, and 2² = 4. Add both to both sides: (x² − 6x + 9) + (y² + 4y + 4) = 12 + 9 + 4.
- Factor: (x − 3)² + (y + 2)² = 25.
- Center (3, −2), radius 5.
This exact question appears on the digital SAT almost every sitting. The SAT math guide lists the other topics that show up most.
Check yourself
The circumference of a circle is 30π. What is its area?
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225π. 2πr = 30π gives r = 15, and π × 15² = 225π.
Find the length of a 135° arc on a circle with radius 8.
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6π ≈ 18.85. (135/360) × 16π = (3/8) × 16π.
A sector has radius 10 and area 25π. What is its central angle in degrees?
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90°. The full circle is 100π, and 25π is a quarter of it.
An inscribed angle intercepts an arc of 140°. What is the angle?
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70°. An inscribed angle is half its arc.
Write the equation of a circle with center (−4, 0) that passes through (2, 8).
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(x + 4)² + y² = 100. The radius is the distance from (−4, 0) to (2, 8): √(36 + 64) = 10.
Find the center and radius of x² + y² + 8x − 2y + 8 = 0.
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Center (−4, 1), radius 3. Completing the square gives (x + 4)² + (y − 1)² = −8 + 16 + 1 = 9.