How to find the Volume of a Triangular Pyramid?

A triangular pyramid is solid with a triangular base and triangles having a shared vertex on all three lateral faces. It’s a tetrahedron with equilateral triangles on each of its four faces. It has a triangle base and four triangular faces, three of which meet at one vertex. A right triangular pyramid’s base is a right-angled triangle, with isosceles triangles on the other faces. All of the faces of a regular triangular pyramid are equilateral triangles and it contains six symmetry planes.

Volume of a Triangular Pyramid

The amount of space occupied by a triangular pyramid in a 3D plane is called its volume. To put it another way, volume specifies the confined area or region of the pyramid. Knowing the base area and height of a triangular pyramid is enough to calculate its volume. Its formula equals one-third the product of base area and height. It is measured in units of cubic meters (m³). 

V = 1/3 × B × h

Where,

V is the volume,

B is the base area,

h is the height of pyramid.

If we are given a regular triangular pyramid consisting of equilateral triangles, its volume is given by the formula,

V = a³/6√2

Where,

V is the volume,

a is the side length.

How to Find the Volume of a Triangular Pyramid?

Let’s take an example to understand how we can calculate the volume of a triangular pyramid.

Example: Calculate the volume of a triangular pyramid of base area 90 sq. m and height 6 m.

Step 1: Note the base area and height of a triangular pyramid. In this example, the base area of the pyramid is 90 sq. m and height is 6 m.

Step 2: We know that the volume of a triangular pyramid is equal to 1/3 × B × h. Substitute the given value of base area and height in the formula.

Step 3: So, the volume of triangular pyramid is calculated as, V = (1/3) × 90 × 6 = 180 cu. m

Sample Problems

Problem 1: Calculate the volume of a triangular pyramid with a base area of 50 sq. m and a height of 4 m.

Solution:

We have,

B = 50

h = 4

Using the formula we get,

V = 1/3 × B × h

= (1/3) × 50 × 4

= 66.67 cu. m

Problem 2: Calculate the volume of a triangular pyramid with a base area of 120 sq. m and a height of 10 m.

Solution:

We have,

B = 50

h = 4

Using the formula we get,

V = 1/3 × B × h

= (1/3) × 120 × 10

= 400 cu. m

Problem 3: Calculate the base area of a triangular pyramid if its volume is 300 cu. m and height is 15 m.

Solution:

V = 300

h = 15

Using the formula we get,

V = 1/3 × B × h

=> B = 3V/h

=> B = 3 (300)/15

=> B = 60 sq. m

Problem 4: Calculate the base area of a triangular pyramid if its volume is 600 cu. m and height are 5 m.

Solution:

V = 600

h = 5

Using the formula we get,

V = 1/3 × B × h

=> B = 3V/h

=> B = 3 (600)/5

=> B = 360 sq. m

Problem 5: Calculate the height of a triangular pyramid if its volume is 200 cu. m and the base area is 60 sq. m.

Solution:

We have,

V = 200

B = 60

Using the formula we get,

V = 1/3 × B × h

=> h = 3V/B

=> h = 3 (200)/60

=> h = 10 m

Problem 6: Calculate the height of a triangular pyramid if its volume is 150 cu. m and the base area is 50 sq. m.

Solution:

We have,

V = 150

B = 50

Using the formula we get,

V = 1/3 × B × h

=> h = 3V/B

=> h = 3 (150)/50

=> h = 9 m

Problem 7: Calculate the volume of a regular triangular pyramid if the side length is 10 m.

Solution:

We have,

a = 10

Using the formula we get,

V = a³/6√2

= (10)³/6√2

= 117.85 cu. m