Equation of a Circle

Understanding Equation of a Circle

The equation comes from the distance formula. Take any point on the curve and compare its horizontal position with the center's horizontal position. Do the same for the vertical positions.

These two changes form the legs of a right triangle. The square of the distance from the center is the horizontal change squared plus the vertical change squared. For a circle, that distance must always be the radius.

Using squares is important because a point can be left or right of the center, above or below it. Squaring makes either direction contribute a positive distance amount.

A common reading error comes from the signs inside the grouped terms. A term written as x minus five means the center is five units to the right. A term written as y plus two means the center is two units down, because y plus two is the same as y minus negative two.

The number on the other side is the radius squared, not the radius itself. If it is sixteen, the radius is four. A reliable graphing method is to mark the center first.

Then mark points one radius up, down, left, and right. These four points give a quick check before drawing the curved boundary.

Completing the square explains how a less organized equation reveals its circle. Consider x squared plus y squared minus six x plus four y minus three equals zero. First move the constant so the two variable groups equal three.

For the x terms, half of negative six is negative three, and its square is nine. For the y terms, half of four is two, and its square is four. Add nine and four to both sides to keep the equation balanced.

Each group now becomes a perfect square. The result has a center at three units right and two units down, with radius four.

Students should write every balancing addition clearly. Adding a value to only one side changes the graph.

Not every equation with squared variables describes a real circle. In the usual coordinate axes, the x squared and y squared terms need matching coefficients after any common factor is removed. A term involving x times y usually signals a rotated conic rather than the simple circle form used in early graphing.

After completing squares, the final radius squared must be positive for an ordinary circle. A value of zero gives just one point. A negative value gives no real points.

These checks matter in coordinate geometry, but they matter in practical work too. A location system may test whether a phone, robot, or game character is within a fixed range of a beacon. Its measured position can be compared with the radius squared, which avoids taking a square root and makes the calculation faster.