Tetrahedron

Introduction to Tetrahedron

A tetrahedron is one of the five platonic solids.

It has triangles as its faces.

A tetrahedron - a solid shape made with 4 triangles


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.

Nets of a tetrahedron folding to a solid shape

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 \(60^\circ\)

A regular tetrahedron - all 4 sides are equilateral triangles

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.

Irregular tetrahedron has triangles as its faces, but they are not equilateral.

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


Tetrahedron Properties

Tetrahedron properties : It consists of 4 faces, 4 corners and 6 edges   

  • 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 \(s\).

Formulas of tetrahedron - A tetrahedron with sides s is shown.

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?

A triangular bypyramid formed by attaching two faces of 2 tetrahedrons. 

Solution:

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

Net of 2 tetrahedrons glued together

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)

A tetrahedron with edge length 5 units

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.

Total surface area of a regular tetrahedron with edge length 6 units 

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 to remember
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.

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

 
 
 
 
 
 
 
Challenge your math skills
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?

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:

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.

There are 5 tetrahedrons in the cube.

  
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