A circle is the set of all points in a plane at a given distance (called the radius ) from a given point (called the center.)
A line segment connecting two points on the circle and going through the center is called a diameter of the circle.
Clearly, if represents the length of a diameter and represents the length of a radius, then .
The circumference of a circle is the distance around the outside. For any circle, this length is related to the radius by the equation
The area of a circle is given the formula
It can be shown that any two circles in the plane are similar , as follows.
Proof that any two circles are similar
Suppose we have two circles, circle centered at with radius and circle centered at with radius .
First, we translate circle A units to the right and units up, so that it is now centered . (Note that and/or may be negative, in which case we are actually shifting the circle left and/or down.)
Then, we perform a dilation, centered at , by a scale factor of . This results in a circle centered at with a radius of .
That is, we have transformed circle into circle , using nothing but translation and dilation. Therefore, the two figures are similar.