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Equation of a Circle

A free Geometry lesson from the “Coordinate Geometry and Proof” unit, with a worked example and practice problems including step-by-step solutions.

A circle in the coordinate plane with center (h, k) and radius r has the equation (x - h)^2 + (y - k)^2 = r^2. This follows directly from the distance formula: every point (x, y) on the circle is exactly r units from (h, k). In Coordinate Geometry and Proof, students need to read the diagram, name the relationship, choose a theorem or formula, and justify why the result follows. The expanded practice now includes fluency, transfer, cumulative review, and proof-style reasoning so Geometry feels connected instead of isolated by topic.

What you'll learn

Why it matters: GPS uses circle equations to find your position from distances to satellites; radar uses them to mark range from a source; and any circular boundary in graphics is described this way.

Worked example

Problem. Write the equation of a circle with center (2, -3) and radius 5.

  1. Use the standard form (x - h)^2 + (y - k)^2 = r^2.
  2. Substitute h = 2, k = -3, r = 5.
  3. (x - 2)^2 + (y - (-3))^2 = 5^2 simplifies to (x - 2)^2 + (y + 3)^2 = 25.
  4. Connect the result back to Equation of a Circle so the geometric relationship is explicit.

Answer: (x - 2)^2 + (y + 3)^2 = 25

Lesson preview: this page shows 5 practice items. The interactive activity contains 28 practice items and 15 quiz items (typed response and multiple choice). Counts include repeated prompts and variants.

Practice problems

1. Circle has center (0, 0) and radius 4. Enter the right side of (x - h)^2 + (y - k)^2 = ?

Show solution
  1. r^2 = 4^2 = 16.

Answer: 16

2. Circle has center (0, 0) and radius 7. Enter the right side.

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  1. r^2 = 49.

Answer: 49

3. Equation (x - 3)^2 + (y - 5)^2 = 25. Enter the radius.

Show solution
  1. r^2 = 25, so r = 5.

Answer: 5

4. Equation (x - 3)^2 + (y - 5)^2 = 25. Enter the x-coordinate of the center.

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  1. h = 3 (from x - 3).

Answer: 3

5. Equation (x + 4)^2 + (y - 1)^2 = 36. Enter the x-coordinate of the center.

Show solution
  1. x + 4 = x - (-4), so h = -4.

Answer: -4

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