Regular Tetrahedron Formula

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

What is Regular Tetrahedron?

A regular tetrahedron is a three-dimensional geometric shape that is a type of polyhedron. It is characterized by having four faces, each of which is an equilateral triangle. In a regular tetrahedron, all the edges have the same length, and all the angles between the faces are equal.

A regular tetrahedron is shown in the image below:

Regular Tetrahedron

Formulas For Regular Tetrahedron

Formulas related to a regular tetrahedron are added below:

Area of One Face of Regular Tetrahedron

For a regular tetrahedron, the area of its one face is given by the formula,

A =[Tex] \frac{\sqrt{3}}{4}x^2 [/Tex]

where x is the side of a Regular Tetrahedron

Slant Height of a Regular Tetrahedron

For a regular tetrahedron, its slant height is given by the formula,

l =[Tex] a(\frac{\sqrt3}{2}) [/Tex]

where a is the Base of Triangle Face

Altitude of a Regular Tetrahedron

For a regular tetrahedron, its altitude is given by the formula,

h = [Tex]\frac{a\sqrt6}{3} [/Tex]

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 = [Tex]\frac{\sqrt{3}}{4}x^2    [/Tex]

 Total Surface Area of Regular Tetrahedron

TSA = 4×[Tex]\frac{\sqrt{3}}{4}x^2[/Tex]

TSA = √3x²

where x is the Length of Side of Regular Tetrahedron

Volume of a Regular Tetrahedron

For a regular tetrahedron, its volume is given by the formula,

V = [Tex]\frac{a^3\sqrt{2}}{12} [/Tex]

where x is the Length of Side of Regular Tetrahedron

Regular Tetrahedron Formulas

Importrant formulas for regular tetrahedron includes:

Area of One Face of Regular Tetrahedron Formula	A = 1/4√(3)a2
Total Surface Area of Regular Tetrahedron Formula	A = 1/4√(3)a2
Volume of a Regular Tetrahedron Formula	V = a3√(2)/12
Slant Height of a Regular Tetrahedron Formula	l = a√(3/2)
Altitude of a Regular Tetrahedron Formula	h = a√(6)/3

Read More:

• Area Formulas
• Tetrahedron Formulas
• Volume Formulas for 3D Shapes

Sample Problems on Regular Tetrahedron Formula

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 = [Tex]\frac{a^3\sqrt{2}}{12} [/Tex]

Here, a = 10 cm

⇒ V =[Tex] \frac{10^3\sqrt{2}}{12} [/Tex]

= 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 =[Tex] \frac{a^3\sqrt{2}}{12} [/Tex]

Here, a = 20 cm

⇒ V =[Tex] \frac{20^3\sqrt{2}}{12} [/Tex]

= 942.809 cm³

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

Solution:

Volume of a tetrahedron =[Tex] \frac{a^3\sqrt{2}}{12} [/Tex]

Here, a = 50 cm

⇒ V = [Tex]\frac{50^3\sqrt{2}}{12} [/Tex]

= 14731.39 cm³

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

Solution:

Volume of a tetrahedron =[Tex] \frac{a^3\sqrt{2}}{12} [/Tex]

Here, a = 40 cm

⇒ V = \frac{40^3\sqrt{2}}{12}

= 7542.47 cm³

Practice Problems on Tetrahedron Formula

Q1. Find the volume of the tetrahedron with edge 23 units.

Q2. If edge of the tetrahedron is 11 units then, find the TSA of tetrahedron.

Q3. Find the edge of the tetrahedron if the volume of tetrahedron given is 252 cubic units.

Q4. Find the edge of the tetrahedron if the TSA of tetrahedron given is 192 square units.

FAQs on Volume of Tetrahedron

What is Formula for a Tetrahedron?

Regular Tetrahedron Formula

Area = √3x² (x is Side Length)

What is Surface Area of a Tetrahedron?

Total Surface Area of Regular Tetrahedron Formula: A = a²√(3).

What is Perimeter of a Tetrahedron?

Perimeter of any regular terahedron with side a is, P = 6a units.

What is Formula for Volume of a Tetrahedron?

Formula for volume of tetrahedron is given by:

Volume of Tetrahedron = a³ / (6√2)