This topic is about circles and the equation of a circle.
Table Of Contents
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 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 where 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 , we have where is the radius (the length from a point on the edge of the circle to the middle of a circle). With as the center of the circle and being a point of the edge of the circle, the equation of the circle is:
In the above equation if and then . This means that the point has zero distance to the center which is also . It can be concluded that is the center of the circle.
If the center of a circle is located at a point , the equation of a circle would be generalized and altered. The general equation of a circle is:
If and then which explains why we have and not with the point as the center of the circle.
The following image below is a visual reference of a circle. The image uses the center while I use . The image is from http://www.mathwarehouse.com/geometry/circle/images/equation-of-circle/general-formula-equation-of-circle.png.
We now look at three cases where a point can be inside the circle, on the circle or outside the circle.
A point is on the circle if .
A point is inside the circle if .
A point is outside the circle if .
Another variation would be using something like but it involves more (unnecessary) work.
What is the radius of a circle centered at with the point on the circle?
We start by using the equation with and and solve for the radius .
Remember that the radius is a distance and has to be at least zero. In this case, we take the positive square root.
Suppose there is a circle centered at with a diameter of 10. Is the point 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 to determine if the left side of the equation is equal to the square of the radius length (which is ). In this equation, we have , , , and .
The point is not on the edge of the circle. Since the square of the radius is greater than 25, the point is outside of the circle centered at with a diameter of 10.
Given a circle centered at the point 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 -values?
Since the diameter length is 4, the radius length is 2. Substituting with into the circle equation gives us:
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 and .