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Equations of Circles

The Hook

The Ring of Fire.

What is it?

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

Time Travel (History)

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

Cheat Code

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

Practice This Now!

The Field Guide

In 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.

In The Real World: 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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Booby Trap!

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.

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

Training Drills

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