Understanding Circles with Inequalities

Instructor: Heidi Decker

Heidi holds a master's degree and has taught secondary mathematics for 17 years.

In this lesson, we will determine if a given point lies inside, outside, or on the boundary of a circle using the equation of a circle in standard form, with the use of inequality symbols.

Location of a Point

Lets say you are looking for a new home to rent in a new city. You want to be able to ride your bike to work so you decide to only look for homes that lie within a 5 mile radius from your new job. How can you determine if any given house is within the 5 mile radius, on the exact circle formed by that 5 mile radius, or farther away than the 5 mile radius?


city map radius


The Boundary Rules

We use inequalities to determine these boundaries, which is a pretty simple concept to follow:

  • If the point (your new home) forms a solution that is exactly EQUAL TO (=) the given radius squared, you know that the point is ON the circle.
  • If the point forms a solution that is GREATER THAN (>) the radius squared, you know that the point lies OUTSIDE the circle.
  • If the point forms a solution that is LESS THAN (<) the radius squared, you know that the point lies INSIDE the circle.

Its that simple!

Circles with Center at the Origin

For example, the standard form of an equation of a circle with its center at the origin (0,0) is x^2 + y^2 = r^2, where the point (x,y) stands for any point that should be on the circle, and r is the radius.

Example 1

Let's look at a specific example, and we'll test several points. Given the equation of a circle with a center at (0,0) and r=5, it would look like: x^2 + y^2 = 5^2 and we can simplify that to:

x^2 + y^2 = 25.

  • Any (x, y) solution that makes x^2 + y^2 = 25 true will be a point on the circle.
  • Any (x, y) solution that makes x^2 + y^2 > 25 true will be a point outside the circle.
  • Any (x, y) solution that makes x^2 + y^2 < 25 true will be a point inside the circle.

Now let's test several points.

Given the point (3, 4), we plug in those values into the equation x^2 + y^2 = 25, and then simplify:

x^2 + y^2 = 25

3^2 + 4^2 = 25

9 + 16 = 25

25 = 25

Since the point (3, 4) gives us a solution (25) that is equal to 25, we can confidently say that the point (3, 4) lies right on the circle.


circle with (3,4)


Try the point (-5, 2). Plug in the values for x and y into the same equation 'x^2 + y^2 = 25, and then simplify:

(-5)^2 + (2)^2 = 25

25 + 4 = 25

29 = 25 ???

Right here we can see that we no longer have an equation, because the two sides are not equal. We need to change this to an inequality symbol:

29 > 25

Since our solution for the point (-5, 2) is GREATER THAN 25, we can say that the point (-5, 2) lies outside the given circle. In the same way, if the solution would have worked out to a number less than 25, we could determine that the point would lie inside the circle.


circle with (-5, 2)

Circles with Center Not at the Origin

When the center of the circle is not at the origin, we have an expanded version of the equation to accommodate the location of the center of the circle: (x - x1)^2 + (y - y1)^2 = r^2 , where (x1, y1) is the center.


moving center

Example 2

For example, if the center is shifted to the right 3 units and down 2 units, we say the center is at C(3, -2). If we keep the radius of 5, our new equation would look like this:

(x - 3)^2 + (y - -2)^2 = 5^2

Simplify: (x - 3)^2 + (y + 2)^2 = 25.

Note that when looking at this new equation, the center point looks like the opposite sign from what you see in the equation; the center is (3, -2), not (-3, 2). Be careful with that!

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