Equation of a Circle
Grade 7The equation
A circle with centre at the origin (0, 0) and radius r has equation:
x² + y² = r²
A circle with centre (a, b) and radius r has equation:
(x − a)² + (y − b)² = r²
This follows directly from Pythagoras — the distance from any point (x, y) on the circle to its centre (a, b) equals the radius.
Reading off the centre and radius
From (x − a)² + (y − b)² = r², the centre is (a, b) and the radius is √(r²). Watch the signs: (x − 3)² means the x-coordinate of the centre is +3, not −3.
The equation (x + 2)² + (y − 5)² = 16 has centre (−2, 5) — the sign flips because the formula has minus signs. Read the centre directly by asking "what makes each bracket zero?"
Worked example
EXAMPLE
A circle has equation (x − 4)² + (y + 1)² = 25. State the centre and radius, and show that the point (8, 2) lies on the circle.
Centre = (4, −1). Radius = √25 = 5. Check (8, 2): (8 − 4)² + (2 + 1)² = 16 + 9 = 25 = r². Centre (4, −1), radius 5; (8, 2) lies on the circle ✓.
