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Equations of Circles

Overview

The Ring of Fire.

What is it?

The algebraic formula that defines every point on a circle based on its center and radius.

History

A specific case of Conic Sections, studied extensively by Apollonius of Perga.

Key Idea

(x-h)² + (y-k)² = r². Center is (h,k).

Practice This Topic

Concept Guide

Plain English: A circle is just a collection of points that are all the exact same distance from a center. The equation is literally the Pythagorean Theorem: (x distance)² + (y distance)² = (radius)².

Real-world example: Seismology. To find an earthquake's epicenter, scientists draw circles around three seismographs. The intersection is the epicenter.

How to do it

  1. Find the center point (h, k).
  2. Find the radius (r).
  3. Square the radius.
  4. Write: (x - h)² + (y - k)² = r².
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Common Pitfall

The Sign Switch. If the equation is (x-3), the center x-coordinate is POSITIVE 3. If it says (x+3), the center is NEGATIVE 3.

Word Problem
"A sprinkler is at coordinates (2, 5) and sprays water 3 meters in every direction. Write the equation for the wet zone."
Reasoning: Center (h,k) = (2,5). Radius r = 3. Equation: (x-2)² + (y-5)² = 9.

Practice Examples

Standard Form
(x-h)² + (y-k)² = r²
Opposite signs for h and k.