The Equation Of A Circle

This topic is about circles and the equation of a circle.


Table Of Contents

  1. Circle Properties
  2. The Equation Of A Circle
  3. Three Cases
  4. Examples


Circle Properties

The circle is a well known shape where every point on the edge of the circle to circle’s center has the same distance. While going through the center of the circle, a straight line from one end of the circle to the other end of the circle is called the diameter. Half the length of this diameter is the radius of a circle. The radius of a circle can be thought of as a straight line from a point on the edge of the circle to the center of the circle.

A point (x,y) is on the circle if the point is on the perimeter of the circle.


The Equation Of A Circle

Recall that from the Pythagorean Theorem, we have the equation a^2 + b^2 = c^2 where c is the longest side or the hypotenuse of a right angled triangle.

The equation of a circle is very similar to the Pythagorean Theorem. Instead of c^2, we have r^2 where r is the radius (the length from a point on the edge of the circle to the middle of a circle). With (0,0) as the center of the circle and (x,y) being a point of the edge of the circle, the equation of the circle is:

    \[x^2 + y^2 = r^2\]

In the above equation if x = 0 and y = 0 then r^2 = 0. This means that the point (0,0) has zero distance to the center which is also (0,0). It can be concluded that (0,0) is the center of the circle.

If the center of a circle is located at a point (p,q), the equation of a circle would be generalized and altered. The general equation of a circle is:

    \[(x - p)^2 + (y - q)^2 = r^2\]

If x = p and y = q then r^2 = 0 which explains why we have (x - p) and not (x + p) with the point (p,q) as the center of the circle.

The following image below is a visual reference of a circle. The image uses the center (h, k) while I use (p,q). The image is from http://www.mathwarehouse.com/geometry/circle/images/equation-of-circle/general-formula-equation-of-circle.png.

 

Equation Of A Circle Source: http://www.mathwarehouse.com/geometry/circle/images/equation-of-circle/general-formula-equation-of-circle

 


Three Cases

We now look at three cases where a point can be inside the circle, on the circle or outside the circle.

A point (p, q) is on the circle if (x - p)^2 + (y - q)^2 = r^2.

A point (p, q) is inside the circle if (x - p)^2 + (y - q)^2 < r^2.

A point (p, q) is outside the circle if (x - p)^2 + (y - q)^2 > r^2.

Another variation would be using something like \sqrt{(x - p)^2 + (y - q)^2} = r but it involves more (unnecessary) work.


Examples

Example One

What is the radius of a circle centered at (0,0) with the point (4, 4) on the circle?

We start by using the equation x^2 + y^2 = r^2 with x = 4 and y = 4 and solve for the radius r.

    \[\begin{array} {lcl} x^2 + y^2 & = & r^2 \ 4^2 + 4^2 & = & r^2 \ 16 + 16 & = & r^2 \ 32 & = & r^2 \ +\sqrt{32} & = & r \ +\sqrt{16 \cdot 2} & = & r \ +\sqrt{16} \cdot \sqrt{2} & = & r \ <ul> <li>4 \sqrt{2} & = & r \</li> </ul> \end{array} \\]

Remember that the radius r is a distance and has to be at least zero. In this case, we take the positive square root.


Example Two

Suppose there is a circle centered at (3, -1) with a diameter of 10. Is the point (2, 5) on the edge of this circle?

The diameter of a circle is twice the radius length. The radius is half the diameter length and would be 5 units.

We use the equation (x - p)^2 + (y - q)^2 = r^2 to determine if the left side of the equation is equal to the square of the radius length (which is 5^2 = 25). In this equation, we have x = 2, y = 5, p = 3, and q = -1.

    \[(x - p)^2 + (y - q)^2 = (2 - 3)^2 + (5 - (-1))^2 = (-1)^2 + 6^2 = 1 + 36 > 25\]

The point (2, 5) is not on the edge of the circle. Since the square of the radius is greater than 25, the point (2, 5) is outside of the circle centered at (3, -1) with a diameter of 10.


Example Three

Given a circle centered at the point (2,2) with a diameter of 4. It is known that we have two points on the circle with an x-value of 2 but with unknown corresponding y-values. What are these corresponding y-values?

Since the diameter length is 4, the radius length is 2. Substituting x = 2 with r = 2 into the circle equation gives us:

    \[\begin{array} {lcl} (x - 2)^2 + (y - 2)^2 & = & 2^2 \ (2 - 2)^2 + (y - 2)^2 & = & 4 \ (0)^2 + (y - 2)^2 & = & 4 \ (y - 2)^2 & = & 4 \ \sqrt{(y - 2)^2} & = & \pm \sqrt{4} \ (y - 2) & = & \pm 2 \ y & = & (\pm 2) + 2 \ y = -2 + 2 \text{ and } y = 2 + 2\ y = 0 \text{ and } y = 4\ \end{array} \\]

We have now found the two points on the circle. These two points on the edge of the circle with an x-value of 2 are (2,0) and (2,4).