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Circles

On the SAT, circle questions usually test formulas, relationships between angles, and equations of circles.

1. Circle Formulas to Know

  • Circumference:
    • c=2\pi r or C=\pi d 
  • Area:
    • A=\pi r^2
  • Arc Length:
    • Arc length = \dfrac{\theta}{360^o} \times 2\pi r 
  • Sector Area:
    • Sector area = \dfrac{\theta}{360^o} \times \pi r^2 

where θ is the central angle in degrees.

2. Angles in Circles

  • Central angle = arc measure.
  • Inscribe angle = half the arc measure.
  • Angles in a semicircle = 90^o.

Example: An inscribed angle intercepts a 100^o arc. The angle = \dfrac{100}{2} = 50^o

3. Equations of Circles

The standard form is:

(x - h) ^2 + (y - k) ^2 = r ^2
  • Center: (h, k)
  • Radius: r

Example:

(x - 3)^2 + (y + 2) = 25
  • Center: (3, -2)
  • Radius = \sqrt{25} = 5

SAT Tips

  1. Find the circumference/area given radius or diameter.
  2. Find arc length/sector area using proportion.
  3. Use inscribed and central angel rules.
  4. Interpret circle equations (find center and radius).
  5. Word problems (tires rotating, clocks, pizza slices, etc.)
  6. Watch units: if diameter is given, remember r = \dfrac{d}{2}.
  7. If you see (x - h)^2 + (y - k)^2 = r^2, don’t forget to take the square root of the constant to find radius.