# Definition:Triangle (Geometry)

*This page is about Triangle in the context of Geometry. For other uses, see Triangle.*

## Definition

A **triangle** is a polygon with exactly three sides.

Thus a **triangle** is a $2$-simplex.

Because it is a polygon, it follows that it also has three vertices and three angles.

## Parts of a Triangle

### Adjacent

The two sides of a triangle which form a particular vertex are referred to as **adjacent** to that angle.

Similarly, the two vertices of a triangle to which a particular side contributes are referred to as **adjacent** to that side.

### Opposite

The side of a triangle which is *not* one of the sides adjacent to a particular vertex is referred to as its **opposite**.

Thus, each vertex has an opposite side, and each side has an opposite vertex.

### Base

For a given triangle, one of the sides can be distinguished as being the **base**.

It is immaterial which is so chosen.

The usual practice is that the triangle is drawn so that the **base** is made horizontal, and at the bottom.

In the above diagram, it would be conventional for the side $AC$ to be identified as the **base**.

### Apex

Having selected one side of a triangle to be the base, the opposite vertex to that base is called the **apex**.

In the above diagram, if $AC$ is taken to be the base of $\triangle ABC$, then $B$ is the **apex**.

### Height

The **height** of a triangle is the length of a perpendicular from the apex to whichever side has been chosen as its base.

That is, the length of the **altitude** so defined.

## Conventional Nomenclature

The vertices of a triangle are conventionally labeled $A, B, C$ (or with other uppercase letters), and the sides with lowercase letters corresponding to the opposite vertex, as above.

In order to emphasize that a particular vertex being referred to is in fact a vertex, the symbol $\angle$ is often placed by the letter corresponding to that vertex.

Thus, for example:

- $\angle A$ is adjacent to sides $b$ and $c$
- Side $a$ is adjacent to $\angle B$ and $\angle C$
- $\angle A$ is opposite side $a$
- Side $a$ is opposite $\angle A$.

## Types of Triangle

### Isosceles Triangle

An **isosceles triangle** is a triangle in which two sides are the same length.

### Equilateral Triangle

An **equilateral triangle** is a triangle in which all three sides are the same length:

That is, a regular polygon with $3$ sides.

### Scalene Triangle

A **scalene triangle** is a triangle in which all three sides are of different lengths.

### Right-Angled Triangle

A **right-angled triangle** is a triangle in which one of the vertices is a right angle.

Note that in order to emphasise the nature of the right angle in such a triangle, a small square is usually drawn inside it.

### Oblique Triangle

An **oblique triangle** is a triangle in which none of the vertices are right angles.

### Acute Triangle

An **acute triangle** is a triangle in which all three of the vertices are acute angles.

### Obtuse Triangle

An **obtuse triangle** is a triangle in which one of the vertices is an obtuse angle.

## Also known as

The word **trigon** can occasionally be seen, but this is rare.

## Also see

- Results about
**triangles**can be found**here**.

## Technical Note

The $\LaTeX$ code to generate $\triangle ABC$ is written `\triangle ABC`

.

Note that `\Delta`

is not to be used, as, although producing a symbol similar in shape, this is actually the uppercase Greek letter delta: $\Delta$

## Euclid's Definition

In the words of Euclid:

**Rectilineal figures**are those which are contained by straight lines,**trilateral**figures being those contained by three,**quadrilateral**those contained by four, and**multi-lateral**those contained by more than four straight lines.

(*The Elements*: Book $\text{I}$: Definition $19$)

## Sources

- 1989: Ephraim J. Borowski and Jonathan M. Borwein:
*Dictionary of Mathematics*... (previous) ... (next):**triangle**(Euclidean geometry) - 1989: Ephraim J. Borowski and Jonathan M. Borwein:
*Dictionary of Mathematics*... (previous) ... (next):**trigon** - 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next):**triangle** - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next):**triangle** - 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next):**triangle** - 2021: Richard Earl and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(6th ed.) ... (previous) ... (next):**triangle**