Circles

Circle

Definition: Circle

Let $\lambda$ be a plane, let $O\in \lambda$ and let $r>0$.

The circle $k$ with radius $r$ and center $O$ is the geometric figure comprised of all points in $\lambda$ which are a distance $r$ from $O$.

NOTATION

$k(O;r)$

Definition: Radius

A radius of the Circle $k(O;r)$ is any line segment whose endpoints are the center $O$ of the circle and a point lying on the $k$.

Theorem

All radii of the same Circle have the same length.

PROOF

TODO

NOTATION

$$rR$$

Definition: Diameter

A diameter of a Circle is a chord which passes through the circle’s center.

Theorem: Length of the Diameter

The length $d$ of every diameter of the Circle $k(O;r)$ is equal to twice the length of $k$‘s radius.

$$d=2r$$

PROOF

TODO

Circumference

Definition: Circumference

The circumference of a Circle is its arc length.

NOTATION

$$C$$

Theorem: Circumference and Radius

The ratio of a Circle’s circumference $C$ to its radius $R$ is always $2\pi$.

$$C=2\pi R$$

PROOF

TODO

Surface Area

Theorem: Surface Area of a Circle

The surface area $S$ of a Circle is given by the length $r$ of its radii as follows:

$$S=\pi r^2$$

PROOF

TODO

Bibliography

1. https://en.wikipedia.org/wiki/Intersecting_chords_theorem