Geometry & Trig · SAT Math

Circles on the Digital SAT

Circle questions on the Digital SAT test a small number of formulas and three big ideas: the equation of a circle, arc-and-sector relationships, and inscribed-angle properties. This lesson covers every formula and pattern you'll need, plus 8 practice problems with full solutions.
By the Prepiii Editorial TeamUpdated 2026-05-23~10 min read

The core circle formulas

Three formulas cover almost every circle question on the SAT:

Circumference: C = 2πr (or πd if you have diameter).

Area: A = πr².

Diameter: d = 2r. The longest distance across the circle.

Memorize these three. The SAT also expects you to know comparable formulas for parts of a circle (arcs and sectors) — see below.

The equation of a circle

A circle on the coordinate plane has the equation:

(x − h)² + (y − k)² = r²

center at (h, k), radius r

Example. (x - 3)² + (y + 2)² = 25 has center (3, -2) and radius 5 (since r² = 25).

Watch the signs. The center coordinates are positive when the equation has (x - h)², and negative when it has (x + h)². The (y + 2)² means k = -2.

Completing the square. If the SAT gives you a messy equation like x² + y² - 6x + 4y - 12 = 0, you need to complete the square in both x and y to find the center and radius. (Or graph it in Desmos — see below.)

Arcs and sectors

Arc length and sector area are fractions of the whole circle, where the fraction equals the central angle divided by 360° (or by 2π in radians).

Arc length: arc = (angle/360) · 2πr. (Fraction of the circumference.)

Sector area: sector = (angle/360) · πr². (Fraction of the area.)

Example. In a circle of radius 6, the central angle is 60°. The arc length is (60/360) · 2π(6) = (1/6) · 12π = 2π. The sector area is (60/360) · π(36) = 6π.

Radians shortcut. If the angle is given in radians (let's call it θ), then arc = rθ and sector area = (1/2)r²θ. Cleaner formulas with no fractions.

Central vs inscribed angles

Two important angle types in circles:

  • Central angle: vertex at the center, arms are two radii. Its measure equals the arc it cuts off.
  • Inscribed angle: vertex on the circle, arms are two chords. Its measure is half the arc it cuts off.

Inscribed angle theorem: An inscribed angle is half the central angle that cuts off the same arc.

Special case: an inscribed angle that cuts off a diameter (a semicircle) is always 90°. The SAT loves this one — it secretly turns the inscribed triangle into a right triangle.

Tangent line property: A tangent line touches the circle at exactly one point, and at that point it is perpendicular to the radius drawn to that point. This often creates a right triangle for Pythagorean work.

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Solving with Desmos

Desmos handles circles natively. Three techniques save real time:

1. Graph the equation to see center and radius

Type (x - 3)² + (y + 2)² = 25. Desmos draws the circle. Hover or click to read off the center and confirm the radius.

2. Skip 'completing the square' by graphing the messy form

For x² + y² - 6x + 4y - 12 = 0, just type it as-is. Desmos plots the circle. Click the center to read the coordinates; the radius is visible from the graph. Saves 60+ seconds of algebra.

3. Compute arc length and sector area with one expression

For arc length: (theta/360) · 2 · pi · r with your values substituted. Desmos returns the exact decimal. Same pattern for sector area.

For the full set of Desmos techniques across the entire test, see our Desmos & Test Tools guides.

Common mistakes

Confusing (x − h) with (x + h) when reading the center

If the equation has (x − 5)², the center's x-coordinate is +5. If it has (x + 5)², the center's x-coordinate is −5. The sign flips.

Forgetting r² is in the equation, not r

(x − 3)² + (y − 2)² = 16 has radius 4, not 16. The right side equals r², so take the square root.

Mixing up arc length and sector area

Arc length is a fraction of circumference (uses 2πr). Sector area is a fraction of area (uses πr²). The two formulas look similar — read which one the question asks for.

Forgetting the inscribed-in-semicircle = right angle rule

If a triangle is inscribed in a circle with one side being the diameter, the angle opposite that diameter is always 90°. Spot this and you instantly have a right triangle.

Practice problems

8 problems adapted from College Board released questions and internal Prepiii sets. Click each one to reveal the solution.

1

A circle has equation (x - 4)² + (y + 3)² = 49. What are the center and radius?

  1. Center (4, 3), radius 49
  2. Center (4, −3), radius 7
  3. Center (−4, 3), radius 7
  4. Center (−4, 3), radius 49

Click to reveal solution →

Answer: (B) Center (4, −3), radius 7

From (x - h)² + (y - k)² = r²: h = 4, k = -3 (sign flip on the +3), r = √49 = 7.

2

A circle has radius 6. What is its area?

  1. 12π
  2. 24π
  3. 36π
  4. 72π

Click to reveal solution →

Answer: (C) 36π

A = πr² = π(6²) = 36π.

3

In a circle of radius 10, a central angle measures 72°. What is the length of the arc it cuts off?

  1. 2π
  2. 4π
  3. 8π
  4. 20π

Click to reveal solution →

Answer: (B) 4π

arc = (72/360) · 2π(10) = (1/5) · 20π = 4π.

4

What is the area of a sector of a circle with radius 4 and central angle 90°?

  1. 2π
  2. 4π
  3. 8π
  4. 16π

Click to reveal solution →

Answer: (B) 4π

sector = (90/360) · π(16) = (1/4) · 16π = 4π.

5

A triangle is inscribed in a circle with one side equal to the diameter (length 10). The other two sides have lengths 6 and 8. What is the area of the triangle?

  1. 24
  2. 30
  3. 40
  4. 48

Click to reveal solution →

Answer: (A) 24

The triangle inscribes a diameter → the opposite angle is 90°. So it's a right triangle with legs 6 and 8 (and hypotenuse 10, confirming 6-8-10 = 3-4-5 scaled).

Area = (1/2) · 6 · 8 = 24.

6

A circle has center (2, -5) and passes through (5, -1). What is its radius?

  1. 3
  2. 4
  3. 5
  4. 7

Click to reveal solution →

Answer: (C) 5

Use the distance formula: r = √((5-2)² + (-1-(-5))²) = √(9 + 16) = √25 = 5.

7

An inscribed angle measures 35°. What is the measure of the arc it cuts off?

  1. 17.5°
  2. 35°
  3. 70°
  4. 145°

Click to reveal solution →

Answer: (C) 70°

The arc is twice the inscribed angle: 2 · 35 = 70°.

8

A circle has circumference 20π. What is its area?

  1. 10π
  2. 20π
  3. 100π
  4. 400π

Click to reveal solution →

Answer: (C) 100π

From C = 2πr = 20π: r = 10.

Area: A = π(10²) = 100π.

Frequently asked questions

What is the equation of a circle?

+
A circle with center (h, k) and radius r has the equation (x − h)² + (y − k)² = r². The signs on h and k flip from what you see inside the parentheses — (x + 3)² means h = −3.

What's the difference between arc length and sector area?

+
Arc length is a portion of the circumference (uses 2πr). Sector area is a portion of the area (uses πr²). Both use the same fraction: angle/360. Read carefully which one the question asks for.

What is the inscribed angle theorem?

+
An inscribed angle (vertex on the circle) is exactly half the central angle that cuts off the same arc. A famous special case: an inscribed angle that subtends a diameter is always 90°, creating a right triangle.

How do I find the center and radius from a messy circle equation?

+
Either complete the square in both x and y to put it into standard form, or just graph it in Desmos and read the center and radius directly. Desmos is usually faster.

What is a tangent line to a circle?

+
A tangent line touches the circle at exactly one point. At that point, the tangent is perpendicular to the radius drawn to it — meaning you can construct a right triangle there.

Keep going

Right triangles

Often appear inscribed in circles

All SAT Math topics

Browse every Digital SAT math topic

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