Tetrahedron

A regular tetrahedron is a three-dimensional geometric shape that is a type of polyhedron.

It is characterized by having

• 4 faces, each of which is an equilateral triangle.
• 4 vertices (corners)
• 6 edges have the same length, and all the angles between the faces are equal.

Formulas

Formulas related to a regular tetrahedron are added below:

Area of One Face

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

A = \frac{\sqrt{3}}{4}x^2

where x is the side of a regular tetrahedron

Slant Height

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

l = a(\frac{\sqrt3}{2})

where a is the Base of Triangle Face

Altitude

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

h = \frac{a\sqrt6}{3}

where a is the Base of Triangle Face

Total Surface Area

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 = \frac{\sqrt{3}}{4}x^2   

 Total Surface Area of Regular Tetrahedron

TSA = 4×\frac{\sqrt{3}}{4}x^2

TSA = √3x²

where x is the Length of Side of Regular Tetrahedron

Volume

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

V = \frac{a^3\sqrt{2}}{12}

where x is the Length of 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 = \frac{a^3\sqrt{2}}{12}

Here, a = 10 cm

⇒ V = \frac{10^3\sqrt{2}}{12}

= 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 = \frac{a^3\sqrt{2}}{12}

Here, a = 20 cm

⇒ V = \frac{20^3\sqrt{2}}{12}

= 942.809 cm³

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

Solution:

Volume of a tetrahedron = \frac{a^3\sqrt{2}}{12}

Here, a = 50 cm

⇒ V = \frac{50^3\sqrt{2}}{12}

= 14731.39 cm³

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

Solution:

Volume of a tetrahedron = \frac{a^3\sqrt{2}}{12}

Here, a = 40 cm

⇒ V = \frac{40^3\sqrt{2}}{12}

= 7542.47 cm³

Practice Problems

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. What is the volume of a regular tetrahedron with an edge length of 10 units?

Q5. What is the surface area of a regular tetrahedron with an edge length of 8 units?