# Definition:Apex

## Definition

The **apex** of a geometric figure is the point which is distinguished from the others by dint of it being furthest away from its base.

Not all figures have a discernible **apex**; for example, parallelograms, prisms and parallelepipeds do not.

### Apex of Triangle

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**.

### Apex of Isosceles Triangle

The vertex opposite the base of an isosceles triangle is called the **apex** of the triangle.

In the above diagram, $A$ is the **apex**.

### Apex of Cone

In the above diagram, the point $A$ is known as the **apex** of the cone.

### Apex of Pyramid

The vertex of a pyramid which is the common vertex of its triangular faces is called the **apex** of the pyramid.

In the above diagram, $Q$ is the **apex**.

## Linguistic Note

The plural of **apex** is **apices**, which is pronounced ** ay-pi-seez**.

The form **apexes** can often be seen, but this is technically incorrect.

Compare vertex.

Hence the colloquial phrase **base over apex** as the description of a particularly flamboyant physical tumble.

## Sources

- 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next): Entry:**apex**(plural**apices**) - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next): Entry:**apex**(plural**apices**) - 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next): Entry:**apex (apices)**