StudyPebble

Circles

Work with circle equations, radius and center, arc length, sector area, and central angles in radians or degrees.

Geometry and TrigonometryMath
Practice circles free

What the SAT actually asks

You handle circles in two settings. In the coordinate plane you read a center and radius out of the standard equation, or complete the square to get there from an expanded form. In plane geometry you compute circumference, area, arc length, sector area, and central or inscribed angles. Arc and sector questions usually come down to taking the fraction of the circle that the central angle covers. Radian measure appears, along with conversions between radians and degrees.

Circle questions typically appear about 1 to 3 times among the 44 math questions, within the roughly 15 percent Geometry and Trigonometry share.

Where students lose points

How to attack these questions

  1. 1Put any circle equation into the standard center-radius form first, completing the square if the equation is expanded.
  2. 2For arcs and sectors, write the fraction of the full circle the angle covers, then multiply it by the circumference or the area.
  3. 3Convert degrees to radians by multiplying by pi over 180, and check whether the answer choices use pi.

Example questions

Written for this skill, with the answer explained. 106 more are in the app.

Try one
Circles
Real question
OAB10
Note: Figure not drawn to scale.

In the figure shown, AB is a diameter of the circle with center O. What is the circumference of the circle?

Medium

In a circle, a central angle measuring 120° intercepts an arc of length 10π centimeters. What is the radius of the circle, in centimeters?

  • A) 10
  • B) 15 ✓
  • C) 30
  • D) 45

Why: A 120 degree angle cuts off 120/360 = 1/3 of the circumference, so (1/3)(2πr) = 10π. Multiplying both sides by 3 gives 2πr = 30π, so r = 15. Answering 30 reports 2r rather than r. Answering 10 reuses the arc's coefficient as the radius, which would give an arc of 20π/3, and a radius of 45 would give an arc of 30π, three times the length given.

Medium

The figure shows two circles with the same center O. What is the area of the region inside the larger circle and outside the smaller circle?

  • A) 16π ✓
  • B) 8π
  • C) 4π
  • D) 2π

Why: Find each area and subtract. The larger circle has area π(5)² = 25π and the smaller has area π(3)² = 9π, so the ring between them is 25π - 9π = 16π. Answering 4π squares the difference of the radii, (5 - 3)², instead of subtracting the squares. Answering 2π uses that difference of radii without squaring at all, and 8π would require the difference of the squares to be 8 rather than 25 - 9.

Harder

Circle O has a radius of 7 inches. What is the length, in inches, of the arc intercepted by a central angle measuring 150 degrees, to the nearest hundredth?

  • A) 9.16
  • B) 18.33 ✓
  • C) 43.98
  • D) 64.14

Why: The arc is 150/360 of the circumference, so its length is (150/360) times 2 pi (7), about 18.33 inches. The value 64.14 is pi times 49 times 150/360, the sector area, which is measured in square inches. Answering 9.16 means you used 7 as the diameter, halving the radius. Answering 43.98 means you gave the circumference of the whole circle.

Questions students ask

What is the equation of a circle?

The standard form is the quantity x minus h squared plus the quantity y minus k squared equals r squared, where the center is at the point with coordinates h and k, and r is the radius. Note that the equation stores the radius squared, so take a square root to get the radius itself.

How do I find arc length on the SAT?

Take the central angle as a fraction of the whole circle, then multiply by the circumference. In degrees that fraction is the angle over 360. In radians the arc length is simply the radius times the angle.

How do I complete the square to find a circle center?

Group the x terms and the y terms, then for each group take half the coefficient of the linear term, square it, and add it to both sides. That turns each group into a perfect square, and the equation is then in standard form.

Drill this skill until it is automatic

Questions on circles alone, with explanations on every one, and the app tracks which ones you keep missing.

Start practicing

Question types inside this skill

Guides that cover this

Other Geometry and Trigonometry skills