SAT Circles

Arc length, circle equations, and radian/degree conversion.

3% of MathMath · Geometry and Trigonometry6 question types
~1per test

How to score it

  • Arc length / circumference = central angle / 360° — set up the proportion.
  • Standard form (x − h)² + (y − k)² = r² gives center (h, k) and radius r directly.
  • If the equation isn't in standard form, complete the square first.
  • Radians → degrees: multiply by 180/π.

Common traps

  • Reads r² as r (forgets the square root).
  • Skips completing the square when the equation is expanded.
  • Confuses the sign of the center coordinates (h, k vs. −h, −k).

The 6 question types, with real examples

Arc length ↔ central angle

“Given a central angle and arc length, what is the [circumference / radius]?”

Practice this →
From the question bankMedium

A circle has a radius of 12. An arc of this circle has a central angle measuring 5π/65\pi /6 radians. What is the length of the arc?

  • A5π5\pi
  • B1010
  • C20π20\pi
  • D10π10\pi✓
Why D

Arc length equals radius times central angle in radians: s=rs = rθ = 12×(5π/6)=10π12 \times (5\pi /6) = 10\pi.

Circle equation (standard form)

“What is the [radius / center] of the circle?”

Practice this →
From the question bankMedium

A circle in the xy-plane has the equation x2+y2−8x+6y=0x^{2} + y^{2} - 8x + 6y = 0.. What is the area of the circle?

  • A5π5\pi
  • B10π10\pi
  • C25π25\pi✓
  • D100π100\pi
Why C

Completing the square: x2x^{2} - 8x8x + y2y^{2} + 6y=06y = 0 becomes (x−4)2−16+(y+3)2−9=0(x - 4)^{2} - 16 + (y + 3)^{2} - 9 = 0, so (x−4)2+(y+3)2=25(x - 4)^{2} + (y + 3)^{2} = 25. The radius is 5, so the area is πr2r^{2} = 25π25\pi.

Radian ↔ degree conversion

“An angle of [fraction]π radians is how many degrees?”

Practice this →
From the question bankMedium

An angle has a measure of 5π/65\pi /6 radians. What is the measure of the angle, in degrees?

  • A7575
  • B100100
  • C150150✓
  • D300300
Why C

To convert radians to degrees, multiply by 180/π180/\pi. Thus, 5π/6×180/π=5×180/6=5×30=1505\pi /6 \times 180/\pi = 5 \times 180/6 = 5 \times 30 = 150 degrees.

Equation that represents a transformed circle

“Circle A has equation [...]. Circle B is the image of A under [translation / scale]. Which equation represents circle B?”

Practice this →
From the question bankMedium

Circle A has equation x2+y2=100x^{2} + y^{2} = 100.. Circle B is the image of circle A under a translation 9 units to the right and 7 units down. Which equation represents circle B?

  • A(x−9)2+(y−7)2=100(x - 9)^{2} + (y - 7)^{2} = 100
  • B(x+9)2+(y+7)2=100(x + 9)^{2} + (y + 7)^{2} = 100
  • Cx2+y2=256x^{2} + y^{2} = 256
  • D(x−9)2+(y+7)2=100(x - 9)^{2} + (y + 7)^{2} = 100✓
Why D

Circle A has center (0, 0) and radius 10. A translation 9 units right and 7 units down moves the center to (0 + 9, 0 - 7) = (9, -7). The radius remains 10, so circle B has equation (x−9)2+(y+7)2=100(x - 9)^{2} + (y + 7)^{2} = 100.

Inscribed / central angle relationships

“[FIGURE: circle with two diameters or an inscribed angle]. What is the [length of arc / measure of angle]?”

Practice this →
From the question bankMedium

In a circle with center O, an inscribed angle intercepts an arc with central angle measure 140 degrees. What is the measure of the inscribed angle, in degrees?

  • A3535
  • B7070✓
  • C140140
  • D280280
Why B

An inscribed angle is half the measure of the central angle that intercepts the same arc. Therefore, the inscribed angle measures 140/2 = 70 degrees.

Sector area from central angle

“A circle has radius [r]. A sector has a central angle of [theta] radians. What is the area of the sector?”

Practice this →
From the question bankMedium

A circle has radius 15. A sector of the circle has a central angle measuring 2π/52\pi /5 radians. What is the area of the sector?

  • A30π30\pi
  • B45π45\pi✓
  • C90π90\pi
  • D7272
Why B

Sector area equals (1/2)r2r^{2}θ = (1/2)(152)(2π/5)=12\displaystyle (15^{2}) (2\pi /5) = \frac{1}{2}(225)(2π/5)=225π/5=45π(2\pi /5) = 225\pi /5 = 45\pi.

Want to try the hard ones? 25 hard Circles questions with full explanations — free, no account needed.

Drill Circles — free trial→

More Geometry and Trigonometry skills

Knowing the pattern is step one. Drilling it is how you score.

SAT Climb turns this into a plan: a diagnostic that finds your weak spots, thousands of calibrated questions across every type, and a mastery grid that shows exactly what to practice next. Free to start.

Start free trial →See all question types

SAT® is a registered trademark of College Board, which is not affiliated with and does not endorse SAT Climb.