Circles
Geometry and Trigonometry is about 15% of SAT Math (5–7 of 44 questions). Circle questions come in two forms: geometry (arcs, sectors, angles, tangent lines, radians) and coordinate geometry (the equation of a circle in the xy-plane, often needing completing the square).
Circumference and area
The circumference is the distance around the circle: C = 2πr = πd. The area is A = πr². Both formulas are on the SAT reference sheet. Notice the difference: the circumference grows with r, the area with r². Many mistakes come from using the diameter where the radius is needed, so halve the diameter first.
| Radius | Circumference 2πr | Area πr² |
|---|---|---|
| 1 | 2π | π |
| 2 | 4π | 4π |
| 3 | 6π | 9π |
| r | 2πr | πr² |
Arcs and sectors
An arc and a sector are fractions of the whole circle. If the central angle is θ degrees, the fraction is θ/360. Arc length = (θ/360) × 2πr. Sector area = (θ/360) × πr². The angle 60° is 1/6 of the circle, 90° is 1/4, and 180° is 1/2.
Central and inscribed angles
A central angle has its vertex at the center, and its measure equals the arc it cuts off. An inscribed angle has its vertex on the circle. It is half of the central angle on the same arc. If two angles cut off the same arc, the inscribed angle is x and the central angle is 2x. An inscribed angle that cuts off a half circle (a diameter) is always 90°.
Radians
A radian is the angle at the center when the arc length equals the radius. A full circle is 2π radians, so 180° = π radians. When θ is in radians, the formulas become simple: arc length s = rθ and sector area = ½r²θ. If the SAT gives an angle in radians, you can use these formulas directly, or turn it into a fraction of 2π (for example, π/3 is 1/6 of 2π).
The equation of a circle
A circle with center (h, k) and radius r has the equation (x − h)² + (y − k)² = r². Watch the signs: (x + 2)² means h = −2. When the equation is written as x² + y² + ax + by = c, complete the square for the x-terms and for the y-terms: add (a/2)² and (b/2)² to both sides. Then read the center and the radius (r² is the number on the right).
Tangent lines
A tangent touches the circle at one point. The radius drawn to that point of tangency is perpendicular to the tangent: they make a right angle. That gives you a right triangle, so you can use the Pythagorean theorem. Two tangents drawn from the same outside point to a circle have equal lengths.
- 4π
- 24π
- 48π
- 144π
- The angle 60° is 60/360 = 1/6 of the full circle.
- Area of the whole circle: π × 12² = 144π.
- Sector area = 144π ÷ 6 = 24π.
- ∠ACB is an inscribed angle that cuts off the same arc AB as the central angle ∠AOB.
- An inscribed angle is half of the central angle on the same arc.
- ∠ACB = 110 ÷ 2 = 55.
- center (−3, 5), radius 4
- center (3, −5), radius 4
- center (3, −5), radius 16
- center (6, −10), radius 4
- Group the terms: (x² − 6x) + (y² + 10y) = −18.
- Complete the square: add (6/2)² = 9 and (10/2)² = 25 to both sides.
- (x² − 6x + 9) + (y² + 10y + 25) = −18 + 9 + 25 = 16.
- (x − 3)² + (y + 5)² = 16, so the center is (3, −5) and r² = 16, r = 4.
- A radius to the point of tangency is perpendicular to the tangent, so triangle OPQ has a right angle at P.
- OQ is the hypotenuse: OP² + PQ² = OQ².
- 25 + PQ² = 169, so PQ² = 144 and PQ = 12.
- This is the 5-12-13 triple.
- Using the diameter as the radiusIn πr² and 2πr use the radius. If the problem gives the diameter or a distance across the circle, halve it.
- Mixing up arc length and sector areaArc length is a length (uses 2πr); sector area is an area (uses πr²). Both multiply by θ/360.
- Sign errors in the circle’s equation(x + 3)² means the center’s x-coordinate is −3. The right-hand side is r², so take the square root.
- Forgetting to add to both sides when completing the squareWhatever you add inside to complete the square, add the same amount on the right side.
- Doubling instead of halving inscribed anglesThe central angle is double the inscribed angle on the same arc, never the reverse.
- Forgetting the right angle at a tangentWhenever a tangent and a radius meet at the point of tangency, draw the right angle and look for the Pythagorean theorem.
Desmos draws a circle from its equation: type (x − 2)^2 + (y + 4)^2 = 9 and the circle appears. You can also type the expanded form, such as x^2 + y^2 − 6x + 10y + 18 = 0, then click the leftmost and rightmost points of the circle. The center lies halfway between them. To check a center, type it as a point such as (3, −5) and see whether it sits in the middle.
- Type (x − 3)^2 + (y + 5)^2 = 16 and look at the circle
- Click the leftmost and rightmost points: the center is halfway between them
- Check an answer choice by typing it and comparing it with the given circle
- Type 144/6 or other arithmetic to evaluate sector areas
Use Desmos to confirm the center and radius when completing the square is messy. Practice completing the square by hand too: it is quick when the coefficients are even.
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