Geometry

Circle Equations

A circle is the set of all points at a fixed distance (the radius) from a center point. On the coordinate plane, this definition translates directly into an equation.

Standard Form of a Circle

The equation of a circle with center (h, k) and radius r is:

(x - h)² + (y - k)² = r²

Use the sliders to move the center and change the radius.

Center h0
-55
Center k0
-55
Radius r3
0.56
(x−0)2+(y−0)2=32(x - 0)^2 + (y - 0)^2 = 3^2

Since FunctionGraph plots y as a function of x, we split the circle into its top half and bottom half by solving for y:

y=0±32−(x−0)2y = 0 \pm \sqrt{ 3^2 - (x - 0)^2 }
-10-8-6-4-2246810-10-8-6-4-2246810Top halfBottom half
Try This

Explore:

  • Move h and k to slide the circle around the plane.
  • Increase r to make the circle bigger.
  • Set h = 0, k = 0, r = 1 — that is the unit circle, one of the most important objects in math.

Why Two Halves?

A circle is not a function — for most x-values, there are two y-values (one above center, one below). So we graph it as two separate functions:

Together they form the complete circle.

Expanded Form and Completing the Square

If you expand (x - h)² + (y - k)² = r², you get:

x² - 2hx + h² + y² - 2ky + k² = r²

or rearranged:

x² + y² - 2hx - 2ky + (h² + k² - r²) = 0

x2+y2−2(0)x−2(0)y+(02+02−32)=0x^2 + y^2 - 2(0)x - 2(0)y + (0^2 + 0^2 - 3^2) = 0

This is the general form. On a test, you might be given an equation like x² + y² + Dx + Ey + F = 0 and asked to find the center and radius. The technique is completing the square.

Connection

Completing the square, step by step:

  1. Group x-terms and y-terms: (x² + Dx) + (y² + Ey) = -F
  2. Complete each square: (x + D/2)² - D²/4 + (y + E/2)² - E²/4 = -F
  3. Rearrange: (x + D/2)² + (y + E/2)² = -F + D²/4 + E²/4
  4. Read off center = (-D/2, -E/2) and r² = -F + D²/4 + E²/4

This is how you convert from general form back to standard form.

Radius and Diameter

The radius r connects the center to any point on the circle. The diameter d = 2r stretches across the full circle through the center.

r=3d=2r=2×3r = 3 \qquad d = 2r = 2 \times 3
0=2πr=2π×3\text0 = 2\pi r = 2\pi \times 3
0=πr2=π×32\text0 = \pi r^2 = \pi \times 3^2

How Far Is a Point from the Circle?

Given a point (x₀, y₀), you can check whether it is inside, on, or outside the circle by evaluating (x₀ - h)² + (y₀ - k)² and comparing it to r²:

Point x₀4
-88
(x0−h)2+(0−k)2=(4−0)2+020r2=32(x_0 - h)^2 + (0 - k)^2 = (4 - 0)^2 + 0^2 \quad \text0 \quad r^2 = 3^2
-10-8-6-4-2246810-10-8-6-4-2246810Top halfBottom halfTest point x₀
Challenge

Challenge: A circle has the equation x² + y² - 6x + 4y - 12 = 0. Use completing the square to find the center and radius. Then set the sliders to verify your answer visually. (Hint: group the x-terms and y-terms separately.)
Take the Quiz