How do you write the equation of a circle?

The standard form of a circle's equation is $(x - h)^2 + (y - k)^2 = r^2$. This formula actually comes directly from the Pythagorean theorem ($a^2 + b^2 = c^2$) and the distance formula. If you draw a right triangle connecting the center $(h, k)$ to any point on the circle $(x, y)$, the horizontal distance is $(x - h)$, the vertical distance is $(y - k)$, and the hypotenuse is the radius $r$. Because every point on the circle is distance $r$ from the center, this relationship always holds true.

Sometimes a problem gives you the center of the circle, but instead of the radius, it gives you another point that lies on the circle. To write the equation, you first need to calculate the radius. You can do this by using the distance formula between the center $(h, k)$ and the point on the circle $(x_1, y_1)$. Once you find that distance, you have your $r$, and you can square it to finish writing your standard form equation.

The most common mistake when writing the equation of a circle is messing up the signs for the center coordinates. Because the formula uses minus signs, $(x - h)$ and $(y - k)$, the signs in the equation will look opposite to the signs of the center's coordinates. For example, if your center is at $(-2, 5)$, the equation will have $(x + 2)^2$ and $(y - 5)^2$. Another frequent error is forgetting to square the radius at the end of the equation. If your radius is 4, the right side of your equation must be 16, not 4.