Equation of a Circle

Key idea

All points at distance r from the centre.

In depth

A circle with centre (h, k) and radius r satisfies (x − h)² + (y − k)² = r². It is the distance formula set equal to r. Expanding and completing the square converts between forms.

Worked example

Centre (2, −1), radius 3: (x − 2)² + (y + 1)² = 9.

Formula

Real-world uses

• Describing a circular coverage area for a Wi-Fi router or a radio mast in mapping software

Common mistake

Sign errors in the centre: (x + 1) means h = −1.

How to learn this

1. Read the key idea and say it in your own words.
2. Work through the worked example, then try a similar problem.
3. Use the interactive lesson’s visual and mini-quiz to check the pattern.

Frequently asked questions

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Related advanced topics in Geometry

• Heron’s Formula – Triangle area from three sides.
• Regular Polygon Area – Use apothem and perimeter.
• Arc Length of a Circle – Part of the circumference.
• Sector Area – Part of the circle’s area.
• Central and Inscribed Angles – An inscribed angle is half the central angle on the same arc.
• Chord and Tangent Properties – A tangent is perpendicular to the radius at the point of contact.