Circle Radius from Centre and a Point

The standard circle equation (x − h)² + (y − k)² = r² says nothing more than "every point on the circle is exactly r away from the centre" — it is the distance formula with the distance held fixed. So the radius of the circle centred at (2, −1) passing through (5, 3) is √(3² + 4²) = 5, the 3-4-5 triangle turning up where it always does. Once you have h, k and r you can write the equation down: (x − 2)² + (y + 1)² = 25.

Watch the signs inside the brackets. A centre at (2, −1) produces (y + 1)², because subtracting a negative adds; reading the equation backwards, (x + 3)² + (y − 4)² = 49 has centre (−3, 4) and radius 7, not (3, −4) and 49. This calculator solves only for the radius, and deliberately so: given r, the centre and one coordinate of the point, a vertical or horizontal line generally cuts the circle in two places, x = h ± √(r² − (y − k)²), so there is no single answer to return. Work those cases by hand and keep both roots — that pair of intersection points is exactly what makes a circle a quadratic curve rather than a linear one. Descartes set all of this in motion in 1637; before coordinates, "the circle through this point about that centre" was a compass instruction, not an equation.