Volume of a regular tetrahedron is a3/(6√2) where a is the edge of the tetrahedron. A pyramid with a triangular base is called a tetrahedron, it is a solid with four triangular faces.
In this article, we will explore how to find the volume of tetrahedrons with solved examples related to the volume of tetrahedrons.
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What is Tetrahedron?
A tetrahedron is a pyramid with a triangular base. It consists of 4 triangles forming a pyramid. In other words, a tetrahedron is a 3-D shape with 4 triangles and 6 edges. The below diagram represents a tetrahedron.
How to Find Volume of a Tetrahedron
To find the volume of a tetrahedron we use the formula of volume of tetrahedron. In this formula we first find the cube of edge of tetrahedron and then divide it by 6√2. The resultant value gives us the volume of the tetrahedron.
Tetrahedron Volume Formula
Formula for the volume of tetrahedron is given by:
Volume of Tetrahedron = a3 / (6√2)
where,
- a is Edge of Tetrahedron
Tetrahedron Volume Formula When Four Points are Given
In this formula we first find three vectors from given four points. Then we apply the formula:
Volume of Tetrahedron = (1/6) × Scalar Product of Three Vectors determined from Given 4 Points
Regular Tetrahedron Formulas
Various Tetrahedron formulas are:
Area of One Face of Regular Tetrahedron Formula | A = 1/4√(3)a2 |
Total Surface Area of Regular Tetrahedron Formula | A = a2√(3) |
Slant Height of a Regular Tetrahedron Formula | l = a√(3/2) |
Altitude of a Regular Tetrahedron Formula | h = a√(6)/3 |
Volume of a Regular Tetrahedron Formula | V = a3√(2)/12 |
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Examples on Volume of Tetrahedron
Example 1: Find the volume of the tetrahedron with edge 6 units.
Solution:
Volume of tetrahedron is given by:
Volume of Tetrahedron = a3 / (6√2)
Volume of tetrahedron = 63 / (6√2)
= 36 / (√2)
= 18√2 cubic units.
Example 2: If edge of the tetrahedron is 4 units then, find the volume of tetrahedron.
Solution:
Volume of tetrahedron is given by:
Volume of Tetrahedron = a3/(6√2)
Volume of tetrahedron = 43 / (6√2)
= 64/ (6√2)
= 32 / (3√2)
= (16√2) / 3 = 7.54 cubic units
Example 3: Find the edge of the tetrahedron if the volume of tetrahedron given is 144√2 cubic units.
Solution:
Volume of tetrahedron is given by:
Volume of Tetrahedron = a3/(6√2)
a3 = Volume of tetrahedron × 6√2
a3 = 144√2 × 6√2
a3 = 1728
a = ∛1728
a = 12 units
Therefore, the edge of given tetrahedron is 12 units.
Practice Problems on Volume of Tetrahedron
1. Find the volume of the tetrahedron with edge 18 units.
2. If edge of the tetrahedron is 9 units then, find the volume of tetrahedron.
3. Find the edge of the tetrahedron if the volume of tetrahedron given is 52 cubic units.
4. Determine the volume of a tetrahedron with an edge length of 10 units.
5. Calculate the volume of a tetrahedron given that its edge length is 15 units.
6. If the volume of a tetrahedron is 80 √2 cubic units, find its edge length.
7. Find the volume of a tetrahedron with an edge length of 3 units.
8. Given a tetrahedron with a volume of 27 cubic units, find the edge length.
9. Calculate the volume of a tetrahedron with an edge length of 12 units.
10. Find the edge length of a tetrahedron if its volume is 54√2 cubic units.
