Tetrahedron
A tetrahedron is a threedimensional shape that has four triangular faces. One of the triangles in a tetrahedron is considered as the base and the other three triangles together form the pyramid. The tetrahedron is one of the types of pyramid, which is a polyhedron with triangular faces connecting the base to a common point and a flat polygon base. A tetrahedron has a triangular base, thus is also referred to as triangular pyramid.
Tetrahedron Definition
A tetrahedron is a polyhedron with 4 faces, 6 edges, and 4 vertices, in which all the faces are triangles. It is also known as a triangular pyramid whose base is also a triangle. A regular tetrahedron has equilateral triangles, therefore, all its interior angles measure 60°. The interior angles of a tetrahedron in each plane add up to 180° as they are triangular.
Net of a Tetrahedron
In geometry, a net can be defined as a twodimensional shape which when folded in a certain manner produces a threedimensional shape. A tetrahedron is a 3D shape that can be formed using a geometric net. Take a sheet of paper. Observe the 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. The folded paper forms a tetrahedron.
Tetrahedron Properties
A tetrahedron is a threedimensional shape that is characterized by some distinct properties. The figure given below shows the face, edge, and vertex of a tetrahedron.
The following points show the properties of a tetrahedron which help us identify the shape easily.
 It has 4 faces, 6 edges, and 4 vertices (corners).
 All four vertices are equidistant from each other.
 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 Formulas
The following table lists the important formulas related to a tetrahedron. Consider a regular tetrahedron made of equilateral triangles with side 's' for the following formulas.
Volume  \(\text{Volume}=\frac{s^3}{6\sqrt{2}}\) 
Total Surface Area  \(\text{TSA}=\sqrt{3} \:s^2 \) 
Area of one face  \(\text{Area of a face }=\frac {\sqrt{3}}{4}s^2 \) 
Slant Height 'l' of a Tetrahedron  \(\text{Slant height}=\frac {\sqrt{3}}{2}s\) 
Altitude 'h' of a Tetrahedron  \(\text{Altitude}=\frac {s\sqrt{6}}{3}\) 
Surface Area of Tetrahedron
The surface area of a tetrahedron is defined as the total area or region covered by all the faces of the shape. The surface area of a tetrahedron is expressed in square units, like m^{2}, cm^{2}, in^{2}, ft^{2}, yd^{2}, etc. A tetrahedron can have two types of surface areas,
 Lateral Surface Area of Tetrahedron
 Total Surface Area of Tetrahedron
Lateral Surface Area of a Tetrahedron
The lateral surface area of a tetrahedron is defined as the surface area of the lateral or the slant faces of a tetrahedron. The formula to calculate the lateral surface area of a regular tetrahedron is given as,
LSA of Regular Tetrahedron = Sum of 3 congruent equilateral triangles(i.e. lateral faces)
LSA of Regular Tetrahedron = 3 ∙ (√3)/4 a^{2} square units
where a is the side length of a regular tetrahedron.
Total Surface Area of a Tetrahedron
The total surface area of a tetrahedron is defined as the surface area of all the faces of a tetrahedron. The formula to calculate the total surface area of a regular tetrahedron is given as,
TSA of Regular Tetrahedron = Sum of 4 congruent equilateral triangles(i.e. lateral faces)
TSA of Regular Tetrahedron = 4 ∙ (√3)/4 a^{2} = √3 a^{2} square units
where a is the side length of the regular tetrahedron.
Volume of Tetrahedron
The volume of a tetrahedron is defined as the total space occupied by a tetrahedron in a threedimensional plane. The formula to calculate the volume of a regular tetrahedron is given as,
Volume of Regular Tetrahedron = (1/3) × area of the base × height = (1/3) ∙ (√3)/4 ∙ a^{2} × (√2)/(√3) a
Volume of Regular Tetrahedron = (√2/12) a^{3} cubic units.
where, a is the side length of the regular tetrahedron.
Topics Related to Tetrahedron
Check out these interesting articles related to the tetrahedron.
Important Notes: The 5 platonic solids can be listed as tetrahedron, cube, octahedron, icosahedron, and dodecahedron.
 A tetrahedron is a triangular pyramid and all 4 faces of a tetrahedron are triangles.
 A tetrahedron has 4 faces, 6 edges, and 4 corners.
Solved Examples on Tetrahedron

Example 1: Two congruent tetrahedrons are stuck together along their base to form a triangular bipyramid. How many faces, edges, and vertices does this bipyramid have?
Solution:
If we open the triangular bipyramid in order to see its net, it will be similar to what is shown in the following figure:
This shows that the triangular bipyramid has 6 triangular faces, 9 edges, and 5 vertices.

Example 2: Find the volume of a regular tetrahedron with a side length measuring 5 units. (Round off the answer to 2 decimal places)
Solution:
We know that the volume of a tetrahedron is:
Volume \(\begin{align}= \frac{s^3}{6\sqrt{2}}\end{align}\), where s = side length.
Substituting 's' as 5 we get:
\[\begin{align}
\text{Volume} &= \frac{5^3}{6\sqrt{2}} \\\\
&=\frac{125}{8.485} \\\\
&\approx 14.73
\end{align}\]
Therefore, the volume of the tetrahedron is 14.73 cubic units. 
Example 3: Each edge of a regular tetrahedron is 6 units. Find its total surface area.
Solution:
The total surface area of a regular tetrahedron is:
Total Surface Area = \(\sqrt{3} \:s^2 \)
Substituting 's' = 6, we get:\[\begin{align}
\text{Total Surface Area} &= \sqrt{3}\times\:6^2 \\\\
&= \sqrt{3} \times 6 \times 6\\
&= 62.35
\end{align}\]
Therefore, the total surface area of the tetrahedron = 62.35 square units.
Practice Questions on Tetrahedron
FAQs on Tetrahedron
What is a Tetrahedron?
A tetrahedron is a platonic solid which has 4 triangular faces, 6 edges, and 4 corners. It is also referred to as a 'Triangular Pyramid' because the base of a tetrahedron is a triangle. A tetrahedron is different from a square pyramid, which has a square base.
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.
 A tetrahedron has 6 planes of symmetry.
Is a Tetrahedron a Pyramid?
Yes, a tetrahedron is a type of pyramid because a pyramid is a polyhedron for which the base is always a polygon and the other lateral faces are triangles. Since a tetrahedron has a triangular base and all its faces are equilateral triangles, it is known as a triangular pyramid.
What is the Difference Between a Tetrahedron and a Triangular Based Pyramid?
A triangular pyramid has its base as a triangle, which may not necessarily be an equilateral triangle, whereas, a tetrahedron is a unique case of a triangular pyramid in which all the faces are equilateral triangles.
Is a Square Based Pyramid a Tetrahedron?
A squarebased pyramid has a square base and all its other faces are triangles, whereas, a tetrahedron has a triangular base and all its faces are equilateral triangles. Thus, a squarebased pyramid is not a tetrahedron.
What is the Base of a Tetrahedron?
A tetrahedron is a figure with 4 triangular faces, therefore, the base of a tetrahedron is also a triangle.
How to Find the Volume of a Tetrahedron?
The volume of a tetrahedron can be calculated by using the formula: \(\frac{s^3}{6\sqrt{2}}\), where 's' is the side length of the tetrahedron. It is measured in cubic units.