Circles

Unit 4 · Geometry & Trigonometry

What the SAT asks here

Circle problems on the SAT focus on three skills: writing the equation from a center and radius, converting the general form to standard form by completing the square, and computing arc length or sector area as a fraction of the whole circle. About one problem per test.

Equation of a circle

Center (h, k), radius r

(x - h)^2 + (y - k)^2 = r^2

General form (convert via completing the square)

x^2 + y^2 + Dx + Ey + F = 0

Arc and sector (central angle θ in degrees)

Circumference / area

C = 2\pi r, \;\; A = \pi r^2

Arc length

s = \frac{\theta}{360^\circ} \cdot 2\pi r

Sector area

A_{\text{sec}} = \frac{\theta}{360^\circ} \cdot \pi r^2

SAT Tip: Arc length and sector area both scale as θ/360° of the full circle. Memorise the fraction, not the formulas — the fraction works in radians too with θ/(2π).

Caution: In (x − h)² + (y − k)² = r², the centre uses the OPPOSITE signs of what appears inside the parentheses. (x + 3)² + (y − 2)² = 25 has centre (−3, 2).

Type 1