Equation of a Circle Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Equation of a Circle.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

The standard form equation (x - h)^2 + (y - k)^2 = r^2 describes a circle with center (h, k) and radius r in the coordinate plane.

A circle is the set of all points at the same distance (the radius) from a center point. The equation just says 'the distance from (x, y) to the center (h, k) equals r,' using the distance formula squared.

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How to Use These Examples

  • Read the first worked example with the solution open so the structure is clear.
  • Try the practice problems before revealing each solution.
  • Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: The circle equation is a direct application of the Pythagorean theorem / distance formula. Every conic section starts from this foundation.

Common stuck point: When converting from general form x^2 + y^2 + Dx + Ey + F = 0, complete the square for both x and y terms separately. Don't forget to add the same constants to both sides.

Sense of Study hint: Group x-terms and y-terms separately, complete the square for each group, then identify the center (h, k) and radius r from standard form.

Worked Examples

Example 1

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

Solution

  1. 1
    The standard form of a circle's equation is (x - h)^2 + (y - k)^2 = r^2, where (h, k) is the center and r is the radius.
  2. 2
    Substitute h = 3, k = -2, r = 5.
  3. 3
    (x - 3)^2 + (y + 2)^2 = 25.

Answer

(x - 3)^2 + (y + 2)^2 = 25
The equation of a circle is derived from the distance formula: every point (x, y) on the circle is exactly r units from the center (h, k). This gives \sqrt{(x-h)^2 + (y-k)^2} = r, which when squared yields the standard form.

Example 2

medium
Find the center and radius of the circle x^2 + y^2 - 6x + 4y - 12 = 0.

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

medium
Find the equation of the circle that has a diameter with endpoints (-1, 4) and (5, -2).

Example 2

hard
Determine whether the circles (x-1)^2 + (y-3)^2 = 16 and (x-7)^2 + (y-3)^2 = 4 intersect, and find the number of intersection points.
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Background Knowledge

These ideas may be useful before you work through the harder examples.

pythagorean theoremdomain