# Geometry

## Circles

### Arc, Chord

By way of introduction, imagine an infinitely thin sheet of paper: it is two dimensional (with two sides. Two points on one side of that surface will be $\small x$ (arbitrary) units apart. If you introduce a small deflection on that surface between those points, the distance between those two points (in two dimensions) will increase. However, if you introduce a third dimension, the points could be represented (e.g.) on the surface of a sphere, where the direct connection between those points in the third dimension (through the volume determined by that topology) is shorter than the distance between those points on the surface of that surface (imagine any two points on a sphere). #### Arc

A section of the circumference of a circle is called an arc. To calculate the length of an arc between points $\small A$ and $\small B$, you need to know the angle at the centre between points $\small A$ and $\small B$. $\small \theta$ (theta) represents the central angle subtended by $\small A$ and $\small B$. You can use degrees or radians for $\small \theta$  [$\small 2 \pi\ rad = 360^\circ$]. You also need to know the radius ($\small r$) of the arc. As there are 360° in the whole circle, the length of the arc is equal to the central angle ($\small \theta$) divided by 360, then multiplied by the circumference of the whole circle ($\small 2 \pi r$):

arc length = $\small 2 \pi r \times (\theta \div 360)$     [$\small \theta$ in degrees].

Example: given $\small r$ = 10 cm, $\small \theta$ = 88°, $\small \pi$ = 3.14159, the arc Length is

2 * 3.14159 * 10 * (88/360) = 15.3589 cm.

If the central angle is in radians (rad), then the arc length is the radius of the arc times the central angle (rad):

arc length = $\small r \theta$     [$\small \theta$ in radians].

There are $\small (2 \pi\ rads)/360^\circ$, so in our previous example), so 88° = 88 x (2 x 3.14159)/360 = 1.535890 rad.

Therefore, the arc length is (10 cm)(1.535890 rad) = 15.3589 cm.

#### Chord

The length of a chord (c) can be calculated using the cosine rule (law of cosines):

chord length $\small c = 2 r\ sin(\theta/2)$     [$\small \theta$ in degrees]

where $\small r$ is the arc radius, and $\small \theta$ is the central angle in degrees. From our earlier example,

chord length c = 2(10)sin(88/2) = (20)sin(44) = 13.8932 cm.

Note that some websites Source
(including Wikipedia Source
!) incorrectly say to use radians in this formula ($\small c = 2 r sin(\theta/2)$), which gives the obviously wrong answer: 0.268 cm!

This website describes it correctly, and here are my handwritten calculations showing how it’s done (note that in my example, I use a central angle of 90°, simplifying the calculations). Property Formula Units (metres shown) Diameter $\small d = 2r$ $\small m$ Circumference $\small C = \pi d = 2 \pi r$ $\small m$ Surface area $\small A = \pi d^2 = 4 \pi r^2$ $\small m^2$ Enclosed volume $\small V = \frac{1}{6} \pi d^3 = \frac{4}{3} \pi r^3$ $\small m^3$