Math Homework. Do It Faster, Learn It Better.


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.

Circle showing radius and diameter

Clearly, if d represents the length of a diameter and r represents the length of a radius, then d = 2 r .

The circumference C of a circle is the distance around the outside. For any circle, this length is related to the radius r by the equation

C = 2 π r

where π (pronounced " pi ") is an irrational constant approximately equal to 3.14 .

The area of a circle is given the formula

A = π r 2 .

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 A centered at ( h 1 , k 1 ) with radius r 1 and circle B centered at ( h 2 , k 2 ) with radius r 2 .

First, we translate circle A h 2 h 1 units to the right and k 2 k 1 units up, so that it is now centered ( h 2 , k 2 ) . (Note that h 2 h 1 and/or k 2 k 1 may be negative, in which case we are actually shifting the circle left and/or down.)

Then, we perform a dilation, centered at ( h 2 , k 2 ) , by a scale factor of r 2 r 1 . This results in a circle centered at ( h 2 , k 2 ) with a radius of r 2 .

That is, we have transformed circle A into circle B , using nothing but translation and dilation. Therefore, the two figures are similar.