Regular Tetrahedron Formula
Last Updated :
09 Aug, 2024
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 a 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 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
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
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:
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
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.
Q5. Calculate the volume of a tetrahedron with an edge length of 15 units.
Q6.Determine the total surface area of a tetrahedron with an edge length of 7 units.
Q7. A tetrahedron has a volume of 1000 cubic units. Find its edge length.
Q8. If the total surface area (TSA) of a tetrahedron is 300 square units, find the edge length.
Q9. What is the volume of a regular tetrahedron with an edge length of 10 units?
Q10. What is the surface area of a regular tetrahedron with an edge length of 8 units?
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