SAT Math

SAT circles: formulas, equations, arcs, and sectors

Learn the circle formulas needed for SAT Math and use worked examples to find radius, area, arc length, sector area, and center.

Written by Luke Finigan

4 min readUpdated Oct 2, 2026

Know $C = 2 π r$ , $A = π r^{2}$ , and $(x - h)^{2} + (y - k)^{2} = r^{2}$ . Arcs and sectors take a fraction of a full circle; use degrees or radians consistently.

The formulas to know and where to find them

Bluebook's reference sheet includes a circle's circumference and area. The coordinate equation, arc-length formulas, and sector formulas are worth learning separately. College Board's reference-sheet example shows what the built-in reference provides.

Start with the radius. A circle with diameter 14 has radius 7, circumference $14 π$ , and area $49 π$ . Using 14 as the radius would double the circumference and quadruple the area.

Keep $π$ exact when the answer choices use it. If the question requests a decimal, round at the end. An area should have square units; a circumference or arc length should have length units.

Read the center and radius from standard form

The equation $(x - 3)^{2} + (y + 2)^{2} = 49$ has center $(3, - 2)$ and radius 7. Standard form is $(x - h)^{2} + (y - k)^{2} = r^{2}$ . The signs inside the parentheses are opposite the corresponding center coordinates.

The right side gives the squared radius. If it is 12, the radius is $12 = 23$ , not 12. If the question asks for the diameter, double that radius.

To check whether $(10, - 2)$ lies on the first circle, substitute: $(10 - 3)^{2} + (- 2 + 2)^{2} = 49$ . It works. Geometrically, the point is seven units to the right of the center, which makes the same check visible.

Circles · Easy

A circle in the

x y

-plane has the equation

(x - 11)^{2} + (y - k)^{2} = 49

. Which of the following gives the center of the circle and its radius?

AThe center is at

(11, k)

and the radius is 7.BThe center is at

(k, 11)

and the radius is 7.CThe center is at

(k, 11)

and the radius is 49.DThe center is at

(11, k)

and the radius is 49.

Choose an answer.

Open the question and explanation

Complete the square when the equation is expanded

Find the center and radius of $x^{2} + y^{2} - 6 x + 4 y - 12 = 0$ . Group the variables and move the constant: $(x^{2} - 6 x) + (y^{2} + 4 y) = 12$ .

For the x group, half of $- 6$ is $- 3$ and its square is 9. For the y group, half of 4 is 2 and its square is 4. Add both squares to both sides: $(x^{2} - 6 x + 9) + (y^{2} + 4 y + 4) = 25$ .

Now factor: $(x - 3)^{2} + (y + 2)^{2} = 25$ . The center is $(3, - 2)$ and the radius is 5. The added constants changed the right side from 12 to 25; forgetting that step gives the wrong radius even if the center is correct.

Use a fraction of a turn for degrees

A $12 0^{\circ}$ central angle spans $120/360 = 1/3$ of a circle. For radius 6, the full circumference is $12 π$ , so its arc length is $4 π$ . The full area is $36 π$ , so its sector area is $12 π$ .

In general, degree-based arc length is $s = \frac{θ}{36 0 ^{\circ}} \cdot 2 π r$ , and sector area is $A = \frac{θ}{36 0 ^{\circ}} \cdot π r^{2}$ . The fraction is the same; the whole-circle quantity changes.

An arc follows the circle's edge. A chord is a straight segment joining two points on it. An arc formula will not give a chord's length. Draw which path the question asks you to measure.

Radians give a shorter formula

A full turn is $2 π$ radians. When $θ$ is in radians, arc length is $s = r θ$ , and sector area is $A = \frac{1}{2} r^{2} θ$ .

For radius 8 and central angle $π /4$ , arc length is $8 (π /4) = 2 π$ and sector area is $\frac{1}{2} (64) (π /4) = 8 π$ . The angle is one eighth of a turn, so you can check both answers against the full circle.

Convert degrees to radians by multiplying by $π /180$ . Thus $6 0^{\circ} = π /3$ . A bare 60 placed into $s = r θ$ would be interpreted as 60 radians, not $6 0^{\circ}$ . Label the angle unit beside the formula.

Use tangents and inscribed angles when they are given

A tangent line meets a circle at one point and is perpendicular to the radius at that point. If an external point is 13 units from the center and the circle's radius is 5, the tangent length is $1 3^{2} - 5^{2} = 12$ .

An inscribed angle is half the measure of its intercepted arc. If it intercepts an arc of $10 0^{\circ}$ , its measure is $5 0^{\circ}$ . A central angle intercepting that arc measures $10 0^{\circ}$ because its vertex is the center.

Practice circles by sorting questions into equations, arc/sector calculations, and angle relationships. If you use Desmos to inspect a circle, confirm the center algebraically and keep exact radicals in the final answer when requested.

Try it yourself

A circle has radius 9 and a central angle of

8 0^{\circ}

. What is the arc length?

2 π

4 π

18 π

20 π

Choose an answer.

Sources

College Board: Bluebook reference-sheet example

Luke Finigan is a student developer and the creator of 1600.now, a free Digital SAT practice platform. About Luke.

01The formulas to know and where to find them

02Read the center and radius from standard form

03Complete the square when the equation is expanded

04Use a fraction of a turn for degrees

05Radians give a shorter formula

06Use tangents and inscribed angles when they are given