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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

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. Warm-up: First identify exactly what the question is asking: Circle has center (0, 0) and radius 4. Enter the right side of (x - h)^2 + (y - k)^2 = ?
  2. For circle problems, connect the formula or theorem to the given radius, diameter, chord, arc, or center information.
  3. r^2 = 4^2 = 16.
  4. Check the result by substituting or estimating: the response should match 16 and make sense in the original problem.
  5. Identify the diagram relationship, formula, theorem, or proof reason before calculating.

Answer: 16

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

Show solution
  1. Warm-up: First identify exactly what the question is asking: Circle has center (0, 0) and radius 7. Enter the right side.
  2. For circle problems, connect the formula or theorem to the given radius, diameter, chord, arc, or center information.
  3. r^2 = 49.
  4. Check the result by substituting or estimating: the response should match 49 and make sense in the original problem.
  5. Identify the diagram relationship, formula, theorem, or proof reason before calculating.

Answer: 49

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

Show solution
  1. Warm-up: First identify exactly what the question is asking: Equation (x - 3)^2 + (y - 5)^2 = 25. Enter the radius.
  2. For circle problems, connect the formula or theorem to the given radius, diameter, chord, arc, or center information.
  3. r^2 = 25, so r = 5.
  4. Check the result by substituting or estimating: the response should match 5 and make sense in the original problem.
  5. Identify the diagram relationship, formula, theorem, or proof reason before calculating.

Answer: 5

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

Show solution
  1. Core Practice: First identify exactly what the question is asking: Equation (x - 3)^2 + (y - 5)^2 = 25. Enter the x-coordinate of the center.
  2. Use inverse operations to isolate the unknown, and keep both sides balanced at every step.
  3. h = 3 (from x - 3).
  4. Check the result by substituting or estimating: the response should match 3 and make sense in the original problem.
  5. Identify the diagram relationship, formula, theorem, or proof reason before calculating.

Answer: 3

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

Show solution
  1. Core Practice: First identify exactly what the question is asking: Equation (x + 4)^2 + (y - 1)^2 = 36. Enter the x-coordinate of the center.
  2. Use inverse operations to isolate the unknown, and keep both sides balanced at every step.
  3. x + 4 = x - (-4), so h = -4.
  4. Check the result by substituting or estimating: the response should match -4 and make sense in the original problem.
  5. Identify the diagram relationship, formula, theorem, or proof reason before calculating.

Answer: -4

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