The Equation of a Circle
“A circle's equation is 'distance r from the center', written with Pythagoras.”
The formula
(x − a)² + (y − b)² = r²How to read it: all points (x, y) at a constant distance r from the center (a, b) form the circle
- (a, b)
- — the coordinates of the center
- r
- — the radius — the distance from center to the circle
- (x, y)
- — any point lying on the circle
The hook
A circle is 'the set of points equally far from one center' — the equation of a circle is just that one sentence written in coordinates.
In plain words
The points (x,y) whose distance from the center (a,b) is always r form the circle. Writing that condition as an equation gives (x−a)²+(y−b)²=r².
The intuition
The distance from the center to a point is the hypotenuse of a right triangle built from the horizontal gap (x−a) and the vertical gap (y−b). By Pythagoras that hypotenuse squared is (x−a)²+(y−b)², and setting it equal to r² says 'the distance is r'.
How it's built
You need just two pieces: the center and the radius. If the center is the origin (0,0), the equation simplifies to x²+y²=r². Watch the signs inside — a center of (2,−3) gives (x−2)²+(y+3)².
Example
A circle centered at (0,0) with radius 5 is x²+y²=25. Plug in (3,4): 9+16=25 → true, so (3,4) lies on this circle.
Common mistake
The right side of (x−a)²+(y−b)²=r² is r², not r. A radius-5 circle has 25 on the right, not 5. And the center's signs appear flipped inside the equation.
Where it's used
GPS positioning (intersecting circles), radar range, collision detection, curves in computer graphics — anywhere 'a fixed distance from a center' matters.
Where it came from
Analytic geometry — treating shapes with coordinates and equations — was opened by Descartes. The equation of a circle is a flagship example of 'solving a geometry problem by calculation'.
Prerequisites
Quick check
What is the equation of a circle centered at (0,0) with radius 3?
- x² + y² = 3
- x² + y² = 6
- x² + y² = 9✓
- (x−3)² + y² = 0
Practice
For a circle centered at (1,2) with radius 4, what is the right side (r²)?
Answer: 16
- The right side is the radius squared
- r² = 4² = 16
Takeaway: The right side is r², not r.
On the circle x²+y²=25, find the positive y at the point (3, y).
Answer: 4
- 3² + y² = 25 → 9 + y² = 25
- y² = 16 → y = 4 (y>0)
Takeaway: Substitute a point to test membership and find coordinates.
For the circle centered at (2,−3) with radius √5, what is the right side of (x−2)²+(y+3)²=?
Answer: 5
- Right side = r² = (√5)²
- = 5
Takeaway: Even an irrational radius can give a clean r².
A circle has (0,0) and (6,0) as the two ends of a diameter. Find its center and radius.
Answer: undefined
- Center is the midpoint: ((0+6)/2, 0) = (3,0)
- Radius is half the diameter: 6/2 = 3
Takeaway: The diameter's midpoint is the center; half the diameter is the radius.