## FANDOM

608 Pages

A triangle is the 2-simplex. The study of triangles is called trigonometry (plus the study of angles). Its Bowers acronym is "trig". It is also known as a pyrogon under the elemental naming scheme.

## Types of Triangle

### By Sides

#### Equilateral

When all of the angles and edges of a triangle are equal a triangle is equilateral. Each angle must have a measure of exactly 60 degrees.

#### Isosceles

When a triangle has two equal sides, it is isosceles. It must also have two equal angles.

#### Scalene

A triangle with three unique side lengths is known as a scalene triangle.

### By Angles

#### Acute

A triangle with all three angles smaller than 90 degrees is called an acute triangle. All equilateral triangles are also acute triangles.

#### Right-Angled

A triangle with one right angle is a right-angled triangle. Their side lengths follow the equation $a^2 + b^2 = c^2$, where c is the edge length of the side opposite the right angle.

#### Obtuse

A triangle where at least one angle is greater than 90 degrees is called an obtuse triangle.

### Special Cases

A triangle with more than one right angle is usually degenerate, but can appear on the surface of a sphere. The same applies to a triangle with angles that sum to greater than 180 degrees, and to triangles that have any angle equal to 180 degrees.

## Structure and Sections

### Hypervolumes

• vertex count = $3$
• edge length = $3l$
• surface area = $\frac{\sqrt{3}}{4} l^2$

### Subfacets

Zeroth First Second Third Fourth Fifth Sixth Seventh Eighth Ninth Tenth Eleventh Twelfth Thirteenth Fourteenth Fifteenth Sixteenth ... Omegath
Hypercube Point Line segment Square Cube Tesseract Penteract Hexeract Hepteract Octeract Enneract Dekeract Hendekeract Dodekeract Tridekeract Tetradekeract Pentadekeract Hexadekeract ... Omegeract
Hyperball Point Line segment Disk Ball Gongol Pentorb Hexorb Heptorb Octorb Enneorb Dekorb Hendekorb Dodekorb Tridekorb Tetradekorb Pentadekorb Hexadekorb ... Omegaball
$\{1\}$ $\{2\}$ $\{3\}$ $\{4\}$ $\{5\}$ $\{6\}$ $\{7\}$ $\{8\}$ $\{9\}$ $\{10\}$ ... $\{\aleph_0\}$
Monogon Digon Triangle Square Pentagon Hexagon Heptagon Octagon Enneagon Decagon ... Apeirogon
Regular
$t_0 \{3\}$
Rectified
$t_1 \{3\}$
Truncated
$t_{0,1} \{3\}$
Triangle Triangle Hexagon