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Circle#

Definition: Circle

Let \(\lambda\) be a [[Planes|plane]], let \(O \in \lambda\) and let \(r \gt 0\).

The circle \(k\) with radius \(r\) and center \(O\) is the [[Geometric Shapes|geometric figure]] comprised of all points in \(\lambda\) which are a distance \(r\) from \(O\).

![[res/Circle.svg]]

Notation

\[k(O;r)\]

Definition: Radius

A radius of the [[Circle]] \(k(O;r)\) is any [[Line Segments|line segment]] whose endpoints are the center \(O\) of the circle and a point lying on the \(k\).

![[res/Radius of a Circle.svg]]

Theorem

All radii of the same [[Circle]] have the same [[Curves|length]].

Proof

TODO

Notation
\[ r \qquad R \]

Definition: Diameter

A diameter of a [[Circle]] is a [[Circle|chord]] which passes through the circle's [[Circle|center]].

![[res/Diameter of a Circle.svg]]

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 [[Curves|arc length]].

Notation
\[ C \]

Theorem: Circumference and Radius

The ratio of a [[Circle]]'s [[Circle#Circumference|circumference]] \(C\) to its [[Circle|radius]] \(R\) is always \(2\pi\).

\[ C = 2\pi R \]
Proof

TODO

Surface Area#

Theorem: Surface Area of a Circle

The [[Surfaces|surface area]] \(S\) of a [[Circle]] is given by the length \(r\) of its [[Circle#Circle|radii]] as follows:

\[ S = \pi r^2 \]
Proof

TODO

Bibliography#

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