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The Circle

By definition, a circle is a set of points on a curve that are a fixed distance, the radius, from a center point.

Deriving The Equation of a Circle

To get the equation of a circle, we need to first go back to our basics -- the pythagorean theorem!

$$(\text{long side})^2 + (\text{short side})^2 = (\text{hypotenuse})^2$$

Imagine the above triangle on a cartesian plane. The horizontal line is $x$ and the vertical line is $y$. The hypotenuse goes from the origin to a point on the circle, designated by $r$, the radius.

We can apply the pythagorean theorem to the above triangle.

$$x^2 + y^2 = r^2$$

Say we want to graph a circle with a radius of $4$. The equation is: $x^2 + y^2 = 4^2$

Applying the Equation of a Circle

So far, we've figured out the equation of a circle that is centered at the origin, $(0,0)$. Let's think about what happens if the circle is not centered at the origin.

Think back to our function transformations!

If the function is not centered at the origin, it will have a different center at $(h,k)$.

So, that means that our new equation for a circle is:

$$(x-h)^2 + (y-k)^2 = r^2$$

Always be careful to watch your signs inside the parentheses! This is your standard form.

For a circle with a center at $(3,2)$ and radius of $5$, our equation is

$$(x-3)^2 + (y-2)^2 = 25$$

Circle Equation

A circle is another type of conic section.

The equation for the standard form of a circle is fairly straightforward, if you consider function transformations.

The standard form of a circle equation is:

$$(x-h)^2 + (y-k)^2 = r^2$$

, where $h$ represents the $x$-coordinate of the center, and $k$ represents the $y$-coordinate of the circle's center. $r$ represents the length of the radius of the circle (distance from the center).

(Video) Circle in Standard Form

by Brian McLogan

In this video by Brian McLogan, you will learn about how to write the standard form of a circle equation, using the center and radius.