Algebra/Chapter 17/Circles

Algebra

Building a Circle

A circle is defined as the set of points that are a fixed distance from a center point. This definition is expressed in the way we draw circles. We pick a point as the center and then use some mechanical means to rotate a drawing utensil around that point. Of course our drawings are always approximations of the shape we think of as a circle. Drawings are only as accurate as the hardness and thickness of our tools will allow. For instance if you blow the drawing below up large enough on your computer screen you might see it is composed of adjacent colored squares and rectangles on your screen. If you used software that maintained the scale of the points as you magnified the circle then eventually it would start to look like a line. This is the reason that we perceive roads as being straight even though we know the Earth is round.

A circle with radii at the integer X and Y coordinates

Still using this drawing We know that the equation for distance is $d={\sqrt {(x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2}}}$. To make the calculations easy let's let point $(x_{1},y_{1})=(0,0)$.

As you likely have learned before, a circle is defined as any set of points that is a fixed distance (the radius) from a fixed point (the center). To find a formula for this, suppose that the center is the point (a,b). According to the distance formula, the distance c from the point (a,b) to any other point (x, y) is:

$c^{2}=(x-a)^{2}+(y-b)^{2}$

By definition, a circle is the set of all points for a given value of c. We call this r and come up with the following equation for a circle:

A circle with center (a, b) and radius r is the set of all points (x, y) such that

$\left(x-a\right)^{2}+\left(y-b\right)^{2}=r^{2}.$