Centre Radius Form of a Circle: Explained with Examples

The centre–radius form of a circle is (x − h)² + (y − k)² = r², where (h, k) is the centre and r is the radius. It is the most readable way to write a circle: you can literally point at the centre and measure the radius straight from the equation, which makes it the form of choice whenever geometry — not algebra — is the focus.

This form matters because it is the bridge between a geometric description ("circle centred at (2, −3) with radius 5") and an equation you can expand, intersect with lines, or feed to a computer. Every circle sketch starts here, and the distance-formula proof behind it is the same proof used for spheres in 3D and for the general conic definitions.

Key Concepts

• Form: (x − h)² + (y − k)² = r². Watch the signs: a centre of (2, −3) gives (x − 2)² + (y + 3)².
• Derivation: it is the distance formula squared — the set of points (x, y) at distance exactly r from (h, k).
• Reading it: complete the square on any x² + y² + 2gx + 2fy + c = 0 to recover this form.
• Diameter form: for endpoints (x₁, y₁) and (x₂, y₂), the circle is (x − x₁)(x − x₂) + (y − y₁)(y − y₂) = 0.

Formulas

• Centre–radius: (x − h)² + (y − k)² = r²; expanded: x² + y² − 2hx − 2ky + h² + k² − r² = 0.
• Radius from general form: r = sqrt(g² + f² − c) with centre (−g, −f).
• Units: h, k and r carry length units; the constant term h² + k² − r² is an area.
• Assumption: r > 0 for a real circle; r = 0 is a point circle.

Worked Example

Write the equation of the circle with centre (2, −3) and radius 5, then check that (−2, −6) lies on it.

1. Substitute into the form: (x − 2)² + (y − (−3))² = 5², i.e. (x − 2)² + (y + 3)² = 25.
2. Expand if needed: x² − 4x + 4 + y² + 6y + 9 = 25, giving x² + y² − 4x + 6y − 12 = 0.
3. Test (−2, −6): (−2 − 2)² + (−6 + 3)² = (−4)² + (−3)² = 16 + 9 = 25. It satisfies the equation, so it lies on the circle.
4. Sanity check on distance: the centre-to-point distance is sqrt((−4)² + (−3)²) = 5, exactly the radius.

Common Misconceptions

• "(x − 2)² + (y + 3)² = 25 has centre (−2, 3)." The centre is (2, −3) — flip the sign of what you see inside each bracket.
• "The radius is 25." 25 is r²; the radius is sqrt(25) = 5.
• "Only integer centres give circles." The centre and radius can be any real numbers, including fractions and irrationals.

Real-World Applications

• Engineering: bolt-hole circles on flanges are specified directly in centre–radius form.
• Game development: circular hitboxes and radar ranges use this form for fast distance checks.
• Geofencing: a circular delivery zone is stored as a centre plus radius in this exact form.
• Optics: spherical mirror and lens surfaces are described by centre–radius equations in ray-tracing software.

In short: (x − h)² + (y − k)² = r² says "all points at distance r from (h, k)" — nothing more. Flip the signs to read the centre, square-root the constant to read the radius, and expand or complete the square to move between this and the general form.