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Original practice · Math

SAT Circles: Find the Center and Radius by Completing the Square

Rewrite a circle equation in standard form, find its center and radius, and verify a point on the circle.

Reviewed

Before you start

Practice circles with an original problem, a complete solution and a transfer check. Try the problem before reading the worked answer.

The original problem

The equation x² + y² − 6x + 4y = 12 represents a circle. What is its radius?

Try the question before reading the solution

Choose one answer, then check it.

Answer: 5

Group the variable terms

Write (x² − 6x) + (y² + 4y) = 12.

Complete both squares

Add 9 and 4 to both sides: (x − 3)² + (y + 2)² = 12 + 9 + 4 = 25.

Read the radius

The standard circle form is (x − h)² + (y − k)² = r². The center is (3, −2), and r² = 25, so r = 5.

A second method or independent check

Expand the proposed form to verify it: (x − 3)² + (y + 2)² = x² − 6x + 9 + y² + 4y + 4. Setting this equal to 25 returns x² + y² − 6x + 4y = 12. The point (8, −2), five units right of the center, also satisfies the original equation: 64 + 4 − 48 − 8 = 12.

Why tempting answers fail

  • 12 is the original right side, which changes when completing squares.
  • 13 is the sum of the two added constants, not the radius.
  • 25 is the square of the radius. Take its positive square root.

Try a transfer problem

What is the radius of x² + y² + 8x − 10y = 8?

Answer: 7

Add 16 and 25: (x + 4)² + (y − 5)² = 49. The radius is √49 = 7.

What to practice next

If this was difficult, return to circles practice and identify the precise step to improve. These are original PeakSAT teaching examples, not official College Board questions or calibrated score predictions. For official adaptive test practice, use Bluebook (opens in a new tab).

Sources

  1. College Board: SAT Math contentsatsuite.collegeboard.org

Sources are linked for reference; they do not endorse PeakSAT. How we prepare our guides