Solving Circle Questions on the SAT

The method: circle equation, complete the square

A circle's standard-form equation, (x − h)² + (y − k)² = r², hands you the center (h, k) and radius r directly. The SAT rarely gives you the equation in that friendly form, though — it usually gives you general form, something like x² + y² + Dx + Ey + F = 0, and expects you to convert.

The tool for that conversion is completing the square, done separately for the x-terms and the y-terms. Group the x's together and the y's together, then for each variable add the square of half its coefficient — and add that same amount to the other side of the equation to keep it balanced. What's left factors into two perfect squares, which is standard form.

Work through a full conversion below, then try one with different numbers.

Arc length and sector area

An arc is a curved piece of the circle's edge; its arc length is a fraction of the full circumference. A sector is the pie-slice region enclosed by two radii and an arc; its area is that same fraction of the full circle's area. In both cases, the fraction is the central angle divided by the full angle of the circle.

With the angle in degrees: Arc length = (θ/360) × 2πr and Sector area = (θ/360) × πr². Every arc-and-sector question is really just "what fraction of the whole circle is this?" — find that fraction first, then multiply by the whole-circle formula.

Central and inscribed angles

A central angle has its vertex at the circle's center; its measure equals the measure of the arc it cuts off. An inscribed angle has its vertex on the circle itself, with both sides as chords; its measure is always exactly half the measure of the arc it intercepts (equivalently, half the central angle that intercepts the same arc).

A special case worth memorizing outright: any inscribed angle that intercepts a semicircle (i.e., its two sides pass through the endpoints of a diameter) measures exactly 90°. That single fact turns a surprising number of circle problems into right-triangle problems.

Formulas to know cold

Equation of a circle: standard form (x − h)² + (y − k)² = r², center (h, k), radius r. General form x² + y² + Dx + Ey + F = 0 — complete the square on both variables to convert.

Circumference and area: C = 2πr = πd and A = πr².

Arc length and sector area: Arc length = (θ/360) × 2πr and Sector area = (θ/360) × πr², where θ is the central angle in degrees. A full circle is 360°, or 2π radians.

Central and inscribed angles: a central angle equals its intercepted arc; an inscribed angle equals half its intercepted arc. An inscribed angle on a diameter is always 90°.

Common traps: arc vs. sector, radius vs. diameter

The most common slip on this topic is reaching for the wrong one of arc length and sector area — they use the same fraction, but arc length is a length (part of the circumference) while sector area is an area (part of the enclosed region). A close second is substituting a diameter where a radius was needed, which throws off every formula by a factor of two.

Before calculating, ask: is the question asking for a distance or a region? Practice that sort below.

Practice questions

Here are five practice problems that help you hone your skills. Use the same method we learned earlier above to solve these problems. Remember, practice makes perfect.