What is The Equation of a Circle and How to Use It

We can graph more than just lines on a coordinate plane! There are equations for many shapes, and a circle is one of them. 

A circle has two major elements. The radius represents the distance 📏 from the circle's center to any point on the circle's circumference, and the center represents the point in the middle of the circle. 

When we work through these problems, we’ll assume the following:

• X1 is the x coordinate of the center of the circle.

• Y1 is the y coordinate of the center of the circle.

• r is the radius of the circle

• A is the area of the circle.

• C is the circumference of the circle.

Standard Form of the Equation of a Circle

The general equation of a circle in standard form is 

This equation allows you to see the center and the radius easily! 

(X1, Y1) is the center of the circle, and the square root of r2 is the radius of the circle. 

Let’s look at three examples that will show you how to write and interpret these equations:

Example 1

Write the equation of the unit circle.

Solution

The unit circle is a circle of radius 1 centered at the origin. We know the center and the radius of the circle, so we can use the standard form of the equation.

Substituting into the equation, we arrive at our solution. We see that the final solution is

Example 2

Write the equation of the circle with radius 5, centered at (8, 10).

Solution

Substituting into the standard form equation, we arrive at the solution

Example 3

Write the equation of the circle with radius 3, centered at (-4, -1).

Solution

Substituting into the equation again, we arrive at the solution.

General Form of the Equation of a Circle

This equation is a rewriting of standard form, but it will often come up in your math problems, so it's essential to know how to deal with it and convert it into the easy-to-use standard form.

The equation of a circle in general form, where A, B, and C are constants, is

You should be able to manipulate this equation to fit the standard form equation, but you won't usually need to write equations in this form.

To transform this equation into the standard form equation, we must complete the square.

To do this, we may follow this step by step guide, which we will illustrate with an example.

Example

Given the equation 

Write the standard form of the equation and determine the center and radius.

Solution

Step 1️⃣: Recall the standard form equation:

Step 2️⃣: From this equation, we can conclude we need to determine X1 and Y1. We can do this by using the values of A and B. Recall that the expansion of (x-a)^2 is x^2 + 2ax + a^2. 

Step 3️⃣: Using this information, we can conclude the values of A and B. In this example, A and B are equal to 5 and 2, respectively.

Step 4️⃣: We must now solve for a^2 in both expansions. We use the values from part 3 to calculate these values.  

From this, we obtain the two values, 25 and 4, that are associated with the values 5 and 2, respectively.

Realize that the sum of these two numbers is greater than the value C in the initial equation, which is 18. We must then add 9 to both sides in order to balance the equation.

Step 5️⃣: Now that we have this information, we may now rewrite the equation such that it is easier to read for the next step:

But wait! We’re not done yet. We must still find the radius and center of the circle. 

If you recall from part 1, once we have the equation in standard form, it is fairly straightforward to determine these values.

Step 7️⃣: We may now determine the center and radius of the circle. We determine that the center is (5,2) and the radius is the square root of 9, or 3.

Takeaways

The SAT and ACT Math sections often test the equation of a circle. You must know how to write the equation and interpret the equation and convert between General Form and Standard Form ✅.

In summary, the standard form of the equation of a circle is 

and the general form of the equation of a circle is 

We can convert from General Form to Standard Form by using a technique called Complete the Square. This technique utilizes the expansion of a perfect square polynomial in order to reduce the equation into its components, two perfect polynomials and a constant term.