Circles

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 $\frac{r_{2}}{r_{1}}$ . 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.

Circles

Proof that any two circles are similar

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