# Tetrahedron

A tetrahedron belongs to the polyhedron family. Based on the edges and angles, tetrahedrons can be regular as well as irregular. This article will precisely deal with the regular tetrahedron.

## What is a Tetrahedron

The word ‘tetrahedron’ came from the Greek word ‘tetráedron’, meaning ‘triangle-based pyramid.’ ‘tetrás’ means ‘four’ and ‘hédra’ means ‘seat’.

A regular tetrahedron is a three-dimensional shape with four faces, six edges, and four vertices. All the faces are equilateral triangles. one of its faces is its base. The other three faces form the pyramid. Thus a tetrahedron is known as a triangular pyramid. It is also a platonic solid – a regular convex polyhedron.

## Properties

1. Has 4 faces; in tetrahedron ABCD, ΔABC, ΔABD, ΔACD, and ΔBCD are the 4 faces
2. Has 6 edges; AB, BC, AC, CD, BD, and AD are the 6 edges
3. Has 4 vertices (corners); A, B, C, D are the 4 vertices
4. Has 4 equilateral triangles as its faces; ΔABC, ΔABD, ΔACD, and ΔBCD are the 4 equilateral triangles
5. All 4 vertices are equidistant from each other; AB = BC = CD = AD
6. Has 6 planes of symmetry
7. Has no parallel faces

## Tetrahedron Net

A tetrahedron net is defined as a two-dimensional shape that produces a three-dimensional figure when folded in a specific manner. We can construct a net with a single sheet of paper cut as in the above figure and then fold it with adhesive to make it a tetrahedron. We call it pyramid packaging. This type of packaging is used for storing liquids such as fruit juice.

## Formulas

### Surface Area

The surface area of a tetrahedron is defined as the total region covered by all its six faces. It is expressed in square units, like sq. m, sq. cm, sq. in, sq. ft, sq. yd, etc. A regular tetrahedron can have two types of surface areas:

1. Total Surface Area – The total surface area of a tetrahedron is defined as the total region covered by all the faces of the shape.

2. Lateral Surface Area – The lateral surface area of a tetrahedron is defined as the surface area of its lateral or the slanted faces of a tetrahedron excluding one face which is the base.

The formulas to calculate the total surface area and  lateral surface area of a tetrahedron are given below:

Let us solve some examples to understand the concept better.

Find the total surface area of a regular tetrahedron with a edge 5 cm.

Solution:

As we know,
Total Surface Area (TSA) = √3a2, here a = 5 cm
= √3 × 52
= 43.30 sq. cm

Find the lateral surface area of a regular tetrahedron with an edge 6 cm.

Solution:

As we know,
Lateral Surface Area (LSA) = (3√3/4) × a2, here a = 6 cm
= (3√3/4) × 62
= 46.76 sq. cm

### Volume

The volume of a tetrahedron is defined as the total space occupied by it in a three-dimensional plane. It is expressed in cubic units such as cm3, m3, in3, ft3, yd3, etc.