The Standard Equation of a Circle
Simple Explanation
A circle with center (h,k) and radius r has the standard equation (x−h)² + (y−k)² = r² — the set of all points at exactly distance r from the center.
Why Do We Need It?
This is the most direct, universally recognized way to describe a circle, and the foundation for recognizing a circle's equation even when written in a less obvious (general) form.
See It
A circle with its center labelled C and a radius line labelled r drawn from the center to the edge of the circle
Formula
The Standard Equation of a Circle
(x−h)² + (y−k)² = r²
A circle is the set of all points at a fixed distance r (the radius) from a fixed center (h,k).
- (h,k)
- — the center of the circle
- r
- — the radius of the circle
When to use it: Whenever a circle needs to be described by an equation, or its center and radius read off directly.
Worked Example
Write a circle's standard equation
Write the equation of a circle with center (3,−2) and radius 4.
Why Does This Work?
The left side, (x−h)²+(y−k)², is exactly the squared distance formula between a point (x,y) and the center (h,k) — setting it equal to r² is precisely the condition "every point on the circle is distance r from the center."
Real-Life Example
Modeling a circular sensor's detection range
A circular motion sensor detects movement anywhere within a fixed radius of its mounted location.
The sensor's detection zone is described exactly by the standard equation of a circle, using its mounted position as the center.
Practice
A circle has equation (x+1)²+(y−5)²=81. Find its radius.
EasyCommon mistake
Forgetting to square the radius on the right side of the equation — writing "=r" instead of "=r²".
Quick Review
- (x−h)² + (y−k)² = r².
- The center is (h,k); the right side is the radius SQUARED.
- Directly built from the distance formula.