Regular Tetrahedron Formula: Definition, Properties, Area & Solved Examples

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Tetrahedron is an interesting three-dimensional figure with four triangular faces and has six straight edges, and four vertex corners. One of the triangles is considered the base and the other three triangles together form a pyramid. For ease of visualization, you can think of it as a triangular pyramid. A tetrahedron is a type of pyramid, a polyhedron with a flat polygonal base and triangular faces connecting the bases and common points. It can be also called a triangular pyramid because its base is a triangle. Elaborated below are the various tetrahedral formulas related to surface area and volume in detail in order to understand the meaning of the symbols used in each formula.

Table of Contents

What is Tetrahedron?
Properties of Tetrahedron
Surface Area of Tetrahedron
Volume of Tetrahedron
Things to Remember
Sample Questions

Key Terms: Tetrahedron, Three-dimensional Figures, Triangle, Pyramid, Polyhedron, Surface Area, Lateral Surface Area, Vertices, Interior Angles

What is Tetrahedron?

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A tetrahedron is a polyhedron with 4 faces, 6 sides, and 4 vertices, all of which are triangles. It can also be called a triangular pyramid whose base is also a triangle. An equilateral tetrahedron has equilateral triangles so all its interior angles are equal to 60°. Since each plane in a tetrahedron is a triangle, thus, the interior angles of a tetrahedron in each plane add up to 180°.

Tetrahedron

Tetrahedron

Properties of Tetrahedron

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A tetrahedron is a three-dimensional shape that features several different properties. The below mentioned are the characteristics of the tetrahedron that make it easy to identify the shape.

There are 4 faces, 6 edges, and 4 vertices (corners).
A regular tetrahedron has all four corners equidistant from each other. There are 6 planes of symmetry.
Unlike other regular polyhedra, there are no parallel faces.
A regular tetrahedron has four equilateral triangles as faces.

Faces and Edges of a Tetrahedron

Faces and Edges of a Tetrahedron

Read More: Area of Equilateral Triangle

Surface Area of Tetrahedron

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The surface area of a tetrahedron is defined as the total area or area covered by all faces of the shape. It is expressed in squares such as m², cm², ft², yd². A tetrahedron has two types of surface area namely the Lateral Surface Area of Tetrahedron and the Total Surface Area of Tetrahedron.

Lateral Surface Area of Tetrahedron

The lateral face of a tetrahedron is defined as the surface of the lateral or slant faces of the tetrahedron. The formula for the area of the sides of a regular tetrahedron is given by,

Lateral Surface Area of Regular Tetrahedron = Sum of 3 congruent equilateral triangles, i.e. lateral faces) = 3 × (√3)/4 a² square units

Here ‘a’ denotes the side length of a regular tetrahedron.

Read More: Edges, Faces, and Vertices

Total Surface Area of Tetrahedron

The total area of a tetrahedron is defined as the area of all the faces of a tetrahedron. The formula for the total area of a regular tetrahedron is given by,

Total Surface Area of Equilateral Quadrilateral = Sum of 4 congruent equilateral triangles (i.e. side faces) = 4 × (√3) / 4 a² = √3 a² square units.

Here ‘a’ denotes the length of the side of the regular tetrahedron

Volume of Tetrahedron

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The volume of a tetrahedron is defined as the total space it occupies in a three-dimensional plane. The formula for the volume of a tetrahedron is given by,

Volume of a regular tetrahedron = (1/3) × base area × height = (1/3) (√3) / 4 a² × (√2) / (√3) a = (√2 / 12) a³ cubic units.

Here ‘a’ is the lengths of the sides of a regular tetrahedron

Tetrahedron Figure

Tetrahedron Figure

Things to Remember

A tetrahedron is a triangular pyramid with all four sides triangular.
Basically, a tetrahedron has 4 faces, 6 edges, and 4 vertex corners.
A tetrahedron can be drawn using a geometric net as it has a 3D shape.
A tetrahedron is made up of 4 triangular faces, that’s why its base is considered a triangle.
Michigan artist David Barr developed the Four Corners project in 1976. A typical Earth-sized tetrahedron crosses the planet with only the four corners protruding at the tips.
The total surface area of a tetrahedron is √3 a² square units while the lateral surface area is 3 × (√3)/4 a² square units.
The volume of a tetrahedron is given by the formula (√2 / 12) a³ cubic units.

Solved Questions

Ques. What is the volume of a tetrahedron with sides of 25 cm? (3 Marks)

Ans. Here, a = 25 cm

Let’s use the formula of Volume of a Regular Tetrahedron: (√2 / 12) a³

= (√2 / 12) 25³

= 1,841.4 cm³

Ques. What is the volume of a Regular Tetrahedron, if its Total Surface Area is 36√3 cm? (5 Marks)

Ans.Here, TSA of Tetrahedron: 36√3 cm

Let's first find out the side of the tetrahedron.

Using the formulae of TSA of Tetrahedron = √3 a²

Now putting the TSA of Tetrahedron = √3 a²

36√3 = √3 a²

36√3 / √3 = a²

36 = a²

Hence a = 6,

Now Using the formulae of Volume of Regular Tetrahedron = (√2 / 12) a³

= (√2 / 12) 63 = 25.45 cm³

Ques. A tetrahedron has four faces and six edges. How many vertices will it have? (3 Marks)

Ans.A tetrahedron will have 4 vertices.

We know that V−E+F=2

Here V, E, F denotes

V= no. of vertices =?

E= no. of edges =6

F= no. of faces =4

V−6+4=2

V−2=2

V=4

Ques. How many vertices and edges does the truncated tetrahedron have? (3 Marks)

Ans.A truncated tetrahedron has 12 vertices and 18 edges in total. A truncated tetrahedron has 4 regular hexagonal faces, 4 equilateral triangle faces, 12 vertices, and 18 edges (of two types). You can construct it by truncating all 4 vertices of a regular tetrahedron at one-third of the original edge length.

Ques. Find out the edge of the tetrahedron if its volume is 1/3? (3 Marks)

Ans.Using the formulae to find out the edge of a tetrahedron,

e = √23√3V

Substituting the volume and solving.

e = √23√3( 1/3)

e = √23√1

e = √2

The edge of the tetrahedron will be √2.

Ques. Find out the edge of the tetrahedron if its volume is 9 cm³? (3 Marks)

Ans.Using the formulae to find out the edge of a tetrahedron,

e = √23√3V

Substituting the volume and solving.

e = √23√3(9)

e = √23√27

e = 3√2

The edge of the tetrahedron will be 3√2.

Ques. Find out the total surface area of a tetrahedron, the edge of a tetrahedron is 12 cm? (3 Marks)

Ans.Using the formulae of the total surface area of tetrahedron= √3a²

Here ‘a’ denotes the edge of the tetrahedron.

= √3(122) = 249.4 cm²

The total surface area of the tetrahedron will be 249.4 cm².

Ques. Find out the lateral surface area of tetrahedron if its volume is 18√3cm³? (3 Marks)

Ans. Let ‘a’ be the edge of the tetrahedron.

Firstly, we will be using the formulae of the volume of tetrahedron = (√2 / 12) a²

18√3 = (√2 / 12) a³ [Putting the given volume of a tetrahedron to find out the edge of the tetrahedron.]

216 = a³

a = 6 cm

Using the formulae of Lateral Surface Area of Tetrahedron = 3 × (√3)/4 a²

= 3 × (√3)/4 (6²)

= 27√3 cm²

The lateral surface area of the tetrahedron will be 27√3cm³.

Ques. The surface area of a regular tetrahedron is 156 cm², find the length of an edge of a regular tetrahedron? (3 Marks)

Ans. Given SA = 156 cm², Let ‘a’ be the length of an edge.

Using the formulae of SA = √3 a²

(Putting the value of SA) 156 = √3 a²

√156√3 = a

a = 9.5 cm

The length of an edge of a regular tetrahedron would be 9.5 cm.

Ques. Find the volume of the regular tetrahedron if its edge is 9.3 m? (3 Marks)

Ans. Given the edge of regular tetrahedron = 9.3 m

Using the formula of Volume of regular tetrahedron = (√2 / 12) a³

=(√2/12) 9.3³

= (0.11785) 9.3³

= 94.79 m³

Related links:

Surface Areas and Volumes	Volume of a Pyramid	Difference Between Cube and Cuboid
Surface Area of Cuboid	Properties of Triangle	Sphere Formula
Types of Polygon	Volume of Cuboid	Area of kite formula
Volume of Sphere	Properties of triangle	Circles

Regular Tetrahedron Formula: Definition, Properties, Area & Solved Examples

What is Tetrahedron?

Properties of Tetrahedron

Surface Area of Tetrahedron

Lateral Surface Area of Tetrahedron

Total Surface Area of Tetrahedron

Volume of Tetrahedron

Things to Remember

Solved Questions

CBSE X Related Questions

1.
PQ and PR are two tangents to a circle with centre O and radius 5 cm. AB is another tangent to the circle at C which lies on OP. If OP = 13 cm, then find the length AB and PA.

2.
Two water taps together can fill a tank in $8\frac{8}{9}$ hours. The tap of larger diameter takes 4 hours less than the smaller one to fill the tank separately. Find the time in which each tap can separately fill the tank.

3.
Use graphical method to solve the system of linear equations : $x = -3$ and $5x - 2y = -5$.

4.
If the zeroes of a polynomial p(x) are $-3$ and 8, then p(x) equals

5.
The dimensions of a window are $156\text{ cm} \times 216\text{ cm}$. Arjun wants to put grill on the window creating complete squares of maximum size. Determine the side length of the square and hence find the number of squares formed.

6.
A bag contains 25 balls. Some of them are yellow and others are green. One ball is drawn at random. If probability of getting a green ball is $3/5$, then find the number of yellow balls.

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Regular Tetrahedron Formula: Definition, Properties, Area & Solved Examples

What is Tetrahedron?

Properties of Tetrahedron

Surface Area of Tetrahedron

Lateral Surface Area of Tetrahedron

Total Surface Area of Tetrahedron

Volume of Tetrahedron

Things to Remember

Solved Questions

CBSE X Related Questions

1.PQ and PR are two tangents to a circle with centre O and radius 5 cm. AB is another tangent to the circle at C which lies on OP. If OP = 13 cm, then find the length AB and PA.

2.Two water taps together can fill a tank in $8\frac{8}{9}$ hours. The tap of larger diameter takes 4 hours less than the smaller one to fill the tank separately. Find the time in which each tap can separately fill the tank.

3.Use graphical method to solve the system of linear equations : $x = -3$ and $5x - 2y = -5$.

4.If the zeroes of a polynomial p(x) are $-3$ and 8, then p(x) equals

5.The dimensions of a window are $156\text{ cm} \times 216\text{ cm}$. Arjun wants to put grill on the window creating complete squares of maximum size. Determine the side length of the square and hence find the number of squares formed.

6.A bag contains 25 balls. Some of them are yellow and others are green. One ball is drawn at random. If probability of getting a green ball is $3/5$, then find the number of yellow balls.

Comments

SUBSCRIBE TO OUR NEWS LETTER

1.
PQ and PR are two tangents to a circle with centre O and radius 5 cm. AB is another tangent to the circle at C which lies on OP. If OP = 13 cm, then find the length AB and PA.

2.
Two water taps together can fill a tank in $8\frac{8}{9}$ hours. The tap of larger diameter takes 4 hours less than the smaller one to fill the tank separately. Find the time in which each tap can separately fill the tank.

3.
Use graphical method to solve the system of linear equations : $x = -3$ and $5x - 2y = -5$.

4.
If the zeroes of a polynomial p(x) are $-3$ and 8, then p(x) equals

5.
The dimensions of a window are $156\text{ cm} \times 216\text{ cm}$. Arjun wants to put grill on the window creating complete squares of maximum size. Determine the side length of the square and hence find the number of squares formed.

6.
A bag contains 25 balls. Some of them are yellow and others are green. One ball is drawn at random. If probability of getting a green ball is $3/5$, then find the number of yellow balls.