Geometry and Trigonometry · Math

Circles

Circles: area and circumference, the standard equation with centre and radius, arcs, sectors, and central angles.

19 questions in the bank · 5 easier · 5 medium · 9 harder · this domain is ≈15% of the section

Learn it in the Guide

What to watch for

The right-hand side of the circle equation is r², not r. For arcs and sectors, the central angle over 360 converts any whole-circle measure to its share.

How the question is worded

These are the actual stems used in the bank — the SAT reuses a small set of wordings, and recognising them saves time on test day.

  • A circle has a radius of 5. What is the circumference of the circle?
  • A circle has a radius of 3. What is the circumference of the circle?
  • A circle has a radius of 7. What is the area of the circle?
  • A circle has a radius of 4. What is the circumference of the circle?

Worked examples

easyA circle has a radius of 5. What is the circumference of the circle?

A circle has a radius of 5. What is the circumference of the circle?

  • A. 5π
  • B. 10π
  • C. 25π
  • D. 50π

Answer: B. 10π

Explanation

Circumference = 2πr = 2π · 5 = 10π. Area (πr²) measures the inside — matching the formula to the quantity asked is the whole question.

mediumThe graph of (x − 2)² + (y + 2)² = 16 in the xy-plane is a circle. What are th…

The graph of (x − 2)² + (y + 2)² = 16 in the xy-plane is a circle. What are the center and radius of the circle?

  • A. Center (−2, 2), radius 4
  • B. Center (2, −2), radius 4
  • C. Center (−2, 2), radius 16
  • D. Center (2, −2), radius 16

Answer: B. Center (2, −2), radius 4

Explanation

The standard form (x − h)² + (y − k)² = r² has center (h, k) and radius r. Matching signs: the equation shows (x − 2), so h = 2; similarly k = −2. The right side is r² = 16, so r = 4, not 16.

hardA circle has radius 6. What is the length of an arc intercepted by a central a…

A circle has radius 6. What is the length of an arc intercepted by a central angle measuring 45°?

  • A. 3/4π
  • B. 9/2π
  • C. 3/2π
  • D. 12π

Answer: C. 3/2π

Explanation

The arc is the fraction 45/360 = 1/8 of the full circumference 2π·6 = 12π: 1/8 × 12π = 3/2π. Central angle over 360 converts any whole-circle measure to its arc or sector share.