**Table of Contents**

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**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:**

\(\text{ Altitude} = \frac {s\sqrt{6}}{3}\) |

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

Enter the edge length in the calculator below.

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**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 units^{3} |

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 units^{2} |

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 units^{2} |

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 \(6\sqrt6\) |

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

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**Practice Questions**

**Here are a few activities for you to practice. **

**Select/Type your answer and click the "Check Answer" button to see the result.**

- 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?
- Rody has a tent which is shaped like a regular tetrahedron. The volume of the tent is 100 m
^{3}and the height is 6 m. What would be the edge length of his tent?

**Maths Olympiad Sample Papers**

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.

You can download the FREE grade-wise sample papers from below:

- IMO Sample Paper Class 1
- IMO Sample Paper Class 2
- IMO Sample Paper Class 3
- IMO Sample Paper Class 4
- IMO Sample Paper Class 5
- IMO Sample Paper Class 6
- IMO Sample Paper Class 7
- IMO Sample Paper Class 8
- IMO Sample Paper Class 9
- IMO Sample Paper Class 10

To know more about the Maths Olympiad you can **click here**

**Frequently Asked Questions(FAQs)**

## 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.