Lesson 10.3.1

Equation of a Circle (Standard Form)

The equation describes every circle on the coordinate plane. The center is and the radius is .

Introduction

Any point (x, y) on the circle is exactly r units from (h, k)

Past Knowledge

Distance formula (1.2.4). Circle vocab (10.1.1). Coordinate plane.

Today's Goal

Write and interpret the standard form equation of a circle.

Future Success

Completing the square (10.3.2), conic sections, SAT problems.

Key Concepts

Standard Form

Center:
Radius: (always positive)

Special Case: Center at Origin

Reading Signs Carefully

→ Center = , . The signs FLIP because of the subtraction in the formula.

Derivation

Deriving the Equation from the Distance Formula

A circle is all points that are exactly units from center .

Distance formula:

Square both sides:

∎ That's it — the equation of a circle IS the distance formula, squared. Every point on the circle satisfies this equation.

Worked Examples

Basic

Write the Equation

Center (2, −3), radius 5. Write the equation.

Intermediate

Center and Point

Center (1, 4), passes through (4, 8). Find the equation.

Find r:

Equation:

Advanced

Is the Point on the Circle?

Circle: . Is (6, 3) on the circle?

Substitute: ✓

Yes — 50 = 50, so the point is on the circle.

Common Pitfalls

Sign Errors

means , not . The formula has subtraction, so a plus sign means the coordinate is negative.

r² vs. r

The equation gives on the right side. If the equation says , then , not 49.

Real-Life Applications

GPS & Trilateration

GPS uses three circle equations (from three satellites) to find your position. Your location is the intersection point of three circles on a coordinate plane.

WiFi & Bluetooth Range

A WiFi hotspot covers a circular area. The equation models a router at the origin with a 50-meter range.