# Tetrahedron

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 1 Introduction to Tetrahedron 2 What is a Tetrahedron? 3 Tetrahedron Properties 4 Tetrahedron Formula 5 Solved Examples on Tetrahedron 6 Important Notes on Tetrahedron 7 Practice Questions on Tetrahedron 8 Challenging Questions on Tetrahedron 10 Maths Olympiad Sample Papers 11 Frequently Asked Questions (FAQs)

## Introduction to Tetrahedron

A tetrahedron is one of the five platonic solids.

It has triangles as its faces.

## What is a Tetrahedron?

A tetrahedron is a three-dimensional shape having all faces as triangles.

### Net of a tetrahedron

Let us do a small activity.

Take a sheet of paper.

You can observe two distinct nets of a tetrahedron shown below.

Copy this on the sheet of paper.

Cut it along the border and fold it as directed in the figure shown below.

The folded paper forms a tetrahedron.

The simulation below illustrates a tetrahedron in 3D.

Click on the edge of the tetrahedron and drag it around.

What do you see?

You will be able to view all the 4 faces of the tetrahedron as it rotates.

A regular tetrahedron has equilateral triangles as its faces.

Since it is made of equilateral triangles, all the internal tetrahedron angles will measure $$60^\circ$$

An irregular tetrahedron also has triangular faces but they are not equilateral.

The internal tetrahedron angles in each plane add up to $$180^\circ$$as they are triangular.

Unless a tetrahedron is specifically mentioned as irregular, by default, all tetrahedrons are assumed to be regular tetrahedrons.

## Tetrahedron Properties

• It has 4 faces, 6 edges and 4 corners.
• All four vertices are equally distant from each other.
• At each of its vertex, 3 edges meet.
• It has 6 planes of symmetry.
• Unlike other platonic solids, a tetrahedron has no parallel faces.
• A regular tetrahedron has equilateral triangles for all its faces.

## Tetrahedron Formula

Various tetrahedron formulas are listed below.

Consider a regular tetrahedron made of equilateral triangles of side $$s$$.

Tetrahedron Volume:

 $$\text{Volume} = \frac{s^3}{6\sqrt{2}}$$

Total Surface Area of a Tetrahedron:

 $$\text{TSA} = \sqrt{3} \:s^2$$

Area of one face of a Tetrahedron:

 $$\text{ Area of a face } = \frac {\sqrt{3}}{4}s^2$$

Slant Height 's' of a Tetrahedron:

 $$\text{ Slant height} = \frac {\sqrt{3}}{2}s$$

Altitude 'h' of a Tetrahedron:

 $$\text{ Slant height} = \frac {s\sqrt{6}}{3}$$

Use the tetrahedron calculator to find the volume and total surface area.

Enter the edge length in the calculator below.

## Solved Examples

 Example 1

Two congruent tetrahedrons are stuck together along its base to form a triangular bipyramid.

How many faces, edges and vertices does this bipyramid have?

Solution:

If we open up the above image to see the net of the triangular bipyramid, we can observe that:

There are 6 triangular faces, 9 edges and 5 vertices.

 Triangular bipyramid has 6 triangular faces, 9 edges and 5 vertices.
 Example 2

Find the volume of a regular tetrahedron with side length measuring 5 units.

(Round off the answer to 2 decimal places)

Solution:

We know that tetrahedron volume whose side $$s$$ is:

\begin{align}\text{Volume} = \frac{s^3}{6\sqrt{2}}\end{align}

Substituting $$s$$ as 5 we get

\begin{align} \text{Volume} &= \frac{5^3}{6\sqrt{2}} \\\\ &=\frac{125}{8.485} \\\\ &\approx 14.73 \end{align}

 Volume of the tetrahedron is 14.73 units3
 Example 3

Each edge of a regular tetrahedron is of length 6 units.

Find its total surface area.

Solution:

The total surface area of a regular tetrahedron of side $$s$$

$$\text{TSA} = \sqrt{3} \:s^2$$

Substituting s = 6,  we get

\begin{align} &\sqrt{3}\times\:6^2 \\\\ &= \sqrt{3} \times 6 \times 6\\ &= 62.35 \end{align}

 Total Surface Area = 62.35 units2
 Example 4

The sum of the length of the edges of a regular tetrahedron is 60 units.

Find the surface area of one of its faces.

Solution:

We know that a regular tetrahedron has 6 edges.

Therefore, the length of each edge is:

\begin{align}\frac{60}{6} = 10 \text{ units}\end{align}

Surface area of one face of the tetrahedron:

\begin{align}\text{ Area of a face } = \frac {\sqrt{3}}{4}s^2 \end{align}

Substituting s = 10, we get:

\begin{align} &\frac{\sqrt{3}}{4}10^2 \\\\ &= \frac{\sqrt{3}}{4} \times 10 \times 10\\ &=25\sqrt{3} \\ &=43.30 \end{align}

 Surface Area of one of its face = 8.66 units2
 Example 5

For what measure of the edge, a tetrahedron's total surface area is equal to its volume?

Solution:

We know that $$\text{TSA} = \sqrt{3} \:s^2$$ and \begin{align}\text{Volume} = \frac{s^3}{6\sqrt{2}}\end{align}

If TSA = Volume, we can say that:

$$\sqrt{3} \:s^2 = \frac{s^3}{6\sqrt{2}}$$

Solving for s, we have

\begin{align} \frac{s^3}{6\sqrt{2}}&= \sqrt{3} \:s^2 \\\\ \frac{s^3}{s^2} &= 6 \times \sqrt{3} \times \sqrt{2}\\ s&=6\sqrt{6} \end{align}

 Edge length of a tetrahedron is $$6\sqrt6$$

Important Notes
1. Tetrahedron, cube, octahedron, icosahedron, and dodecahedron are the only 5 platonic solids.
2. A tetrahedron is a triangular pyramid; all 4 faces of a tetrahedron are triangles.
3. A tetrahedron has 4 faces, 6 edges and 4 corners.

## Practice Questions

Here are a few activities for you to practice.

Challenging Questions
1.  A new shape is formed by aligning the face of a tetrahedron exactly over one triangular face of the square pyramid. How many vertices, edges and faces will the new shape have?
2. Rody has a tent which is shaped like a regular tetrahedron. The volume of the tent is 100 m3 and the height is 6 m. What would be the edge length of his tent?

IMO (International Maths Olympiad) is a competitive exam in Mathematics conducted annually for school students. It encourages children to develop their math solving skills from a competition perspective.

## 1. What is tetrahedron?

A tetrahedron is a platonic solid having triangles as its faces.

## 2. What are the properties of a tetrahedron?

The properties of a tetrahedron are:

• It has 4 faces, 6 edges and 4 corners.
• All four vertices are equally distant from each other.
• Unlike other platonic solids, tetrahedron has no parallel faces.
• A regular tetrahedron has all its faces as equilateral triangles.
• At each vertex of a tetrahedron, 3 edges meet.
• A tetrahedron has 6 planes of symmetry.

## 3. How many tetrahedrons are in a cube?

There are 5 tetrahedrons in a cube.

The centre tetrahedron is regular and the others are irregular.

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