Equation of a Circle and the Pythagorean Theorem

The equation of a circle with its center at the origin and radius r is x² + y² = r². It comes straight from the Pythagorean theorem: for any point (x, y) on the circle, the horizontal distance x and the vertical distance y are the legs of a right triangle, and the radius r is its hypotenuse.

Move the center to a point (h, k) and the same reasoning gives the equation of a circle with any center, called the center-radius form or standard form:

Deriving x² + y² = r² from the Pythagorean theorem

A circle is the set of all points that are the same distance, the radius r, from a fixed point, the center. Put the center at the origin O(0, 0) and pick any point P(x, y) on the circle. Drop a vertical line from P to the x-axis, meeting it at F(x, 0).

Triangle OFP is a right triangle with the right angle at F, because the vertical line meets the horizontal axis at 90°. Its legs are OF, with length |x|, and FP, with length |y|. Its hypotenuse is OP, which is a radius, so its length is r. The Pythagorean theorem says:

The absolute-value bars disappear because a square is never negative: (−3)² = 3² = 9. That is why one equation covers all four quadrants. Watch the point travel round the circle below; the triangle changes shape at every moment, but its hypotenuse is always the radius.

When P sits on an axis, say at (5, 0), the triangle flattens into a segment, but the equation still holds: 5² + 0² = 25.

The argument also runs backward. If a point satisfies x² + y² = r², then its distance from the origin is √x² + y² = r, so it lies on the circle. Points on the circle satisfy the equation and points that satisfy the equation are on the circle, so the equation describes the circle exactly, with nothing missing and nothing extra.

The center-radius form (x − h)² + (y − k)² = r²

Now let the center be any point C(h, k). The right triangle is still there, but its legs are measured from the center instead of from the origin. The horizontal leg is x − h and the vertical leg is y − k (up to sign), so the theorem gives

This is the distance formula from (h, k) to (x, y), set equal to r and squared to clear the square root. The circle centered at the origin is the special case h = k = 0.

Worked examples

From the general form to the center-radius form

If you expand the center-radius form and move everything to one side, you get the general form x² + y² + Dx + Ey + F = 0. It hides the center and radius, so to read them you complete the square: take half of each x and y coefficient, square it and add it to both sides.

Is a point inside, on or outside the circle?

The left side of the equation, (x − h)² + (y − k)², is the square of the distance from the center to the point (x, y). Compare it with r²:

• (x − h)² + (y − k)² < r²: the point is inside the circle.
• (x − h)² + (y − k)² = r²: the point is on the circle.
• (x − h)² + (y − k)² > r²: the point is outside the circle.

You never need a square root. Distances are positive, and for positive numbers a bigger square always means a bigger number, so comparing squares tells you the same thing as comparing distances.

The same "compare the squares" idea classifies triangles. Draw the circle that has a segment AB as its diameter. For any point P not on the line AB, the angle APB is a right angle if P is on the circle, obtuse if P is inside and acute if P is outside. The acute, right and obtuse triangles page turns this into the test c² versus a² + b².

Solving for y: why a circle is not a function

Solving x² + y² = r² for y gives two answers:

The plus sign gives the top half of the circle and the minus sign the bottom half. On x² + y² = 25, the value x = 3 gives y = 4 and y = −4. Because most x-values have two y-values, a circle fails the vertical line test and is not the graph of a function. To draw one on a graphing calculator, enter the two halves, y = √25 − x² and y = −√25 − x², as separate functions. When |x| > r, the number under the root is negative, which is the algebra's way of saying the vertical line misses the circle.

The unit circle and sin²θ + cos²θ = 1

The circle of radius 1 centered at the origin is the unit circle, x² + y² = 1. In trigonometry, the point on it at angle θ from the positive x-axis has coordinates (cosθ, sinθ). Substituting into the circle's equation gives

This is the Pythagorean identity: the Pythagorean theorem, the equation of a circle and the most-used trigonometric identity are one fact seen three ways.

Pythagorean triples as points on a circle

Points on a circle whose coordinates are both whole numbers are called lattice points. The circle x² + y² = 25 passes through exactly 12 of them. Four lie on the axes; the other eight come from the single Pythagorean triple 3, 4, 5, reflected into every quadrant and swapped between x and y.

Every off-axis lattice point on a circle x² + y² = r² with whole-number r gives a triple (|x|, |y|, r). The circle x² + y² = 169 has 12 lattice points, including (5, 12), from the 5-12-13 triangle. The circle x² + y² = 625 has 20, because 25 is the hypotenuse of two different triples, 7-24-25 and 15-20-25.