The equation of a circle

A circle is the set of all points that are the same distance from one fixed point. That fixed point is the center, and the shared distance is the radius. Here we take a circle with center at the origin $(0, 0)$ and radius $5$.

Pick any point $(x, y)$ on the circle. Its distance to the center is always $5$. Measured from the center it is $x$ across and $y$ up, which makes a right triangle, so by the Pythagorean theorem the horizontal side squared plus the vertical side squared equals the slanted side (the radius) squared: $x^2 + y^2 = 5^2$. Since $5^2 = 25$, the equation of this circle is $x^2 + y^2 = 25$.

Let us check with the point $(3, 4)$: $3^2 + 4^2 = 9 + 16 = 25$, which works, so $(3, 4)$ lies on the circle. In the same way $(5, 0)$, $(0, 5)$ and $(-4, 3)$ all give $25$, so they are on the circle too. On the other hand $(1, 1)$ gives $1^2 + 1^2 = 2$, not $25$, so that point is inside the circle, not on it. So you can test whether a point is on the circle just by checking whether it satisfies the equation.

The idea is the same for any radius $r$: a circle centered at the origin is $x^2 + y^2 = r^2$. If the center moves to $(a, b)$, the horizontal gap becomes $x - a$ and the vertical gap $y - b$, so the equation becomes $(x - a)^2 + (y - b)^2 = r^2$. This is the general equation of a circle.

The graph is drawn by solving the equation for $y$. Solving $x^2 + y^2 = 25$ gives $y = \sqrt{25 - x^2}$ (the top half) and $y = -\sqrt{25 - x^2}$ (the bottom half), and putting them together makes the whole circle. The two halves join at the ends $(5, 0)$ and $(-5, 0)$, where $y = 0$, so together they form one unbroken circle. Because one value of $x$ has two matching values of $y$, a circle is not a function of the form $y = f(x)$, which is why it is drawn as two separate curves.

The large dots on the graph are the center $(0, 0)$ and the point $(3, 4)$ on the circle. No matter which direction you go from the center, the distance out to the circle is always the radius $5$ — and that is exactly what the equation of a circle says.