Regular Tetrahedron Formula

A regular tetrahedron is a three-dimensional figure composed of four triangular faces, each of which is equilateral. All four faces are congruent to each other. It can also be referred to as a pyramid which is triangular in shape. A tetrahedron has 4 faces, 6 edges, and 4 vertices. It is shown in the figure below.

Formulas For Regular Tetrahedron

Area of One Face of Regular Tetrahedron

A =

where, x is side of regular tertahedron.

Slant Height of a Regular Tetrahedron

h =

where, a is the base of triangle face.

The altitude of a Regular Tetrahedron

h = 

where, a is the base of triangle face.

Total Surface Area of a Regular Tetrahedron Formula

Since a regular tetrahedron is composed of four equilateral triangles, naturally its surface area would be the sum total of the areas of all those equilateral triangles. Now, the area of an equilateral triangle with the side x is

 Area of equilateral triangle = 

 Total surface area of  regular tetrahedron TSA = 4*

                                                                          = √3x²

where x is the length of the side of regular tetrahedron.

Volume 

V =

where x is the length of the side of regular tetrahedron.

Sample Problems

Problem 1: Calculate the TSA of a tetrahedron of side 4 cm.

Solution:

TSA of a tetrahedron = √3x²

Here, x = 4 cm

⇒ TSA = √3x (4)²

= 27.712 cm²

Problem 2: Calculate the volume of a tetrahedron of the side 10 cm.

Solution:

Volume of a tetrahedron = 

Here, a = 10 cm

⇒ V =

= 117.85 cm³

Problem 3: Calculate the TSA of a tetrahedron of the side 10 cm.

Solution:

TSA of a tetrahedron = √3x²

Here, x = 10 cm

⇒ TSA = √3 x (10)²

= 173.20 cm²

Problem 4: Calculate the TSA of a tetrahedron of the side 30 cm.

Solution:

TSA of a tetrahedron = √3x²

Here, x = 30 cm

⇒ TSA = √3 x (30)²

= 1558.84 cm²

Problem 5: Calculate the volume of a tetrahedron of the side 20 cm.

Solution:

Volume of a tetrahedron =

Here, a = 20 cm

⇒ V =

= 942.809 cm³

Problem 6: Calculate the volume of a tetrahedron of the side 50 cm.

Solution:

Volume of a tetrahedron =

Here, a = 50 cm

⇒ V = 

= 14731.39 cm³

Problem 7: Calculate the volume of a tetrahedron of the side 40 cm.

Solution:

Volume of a tetrahedron =

Here, a = 40 cm

⇒ V = \frac{40^3\sqrt{2}}{12}

= 7542.47 cm³