In geometry, a pyramid is a polyhedron formed by connecting a polygonal base and a point (vertex). Pyramids are generally classified by their bases. The pyramids have square bases, are called square pyramids, a hexagonal pyramid has a base that is a hexagon and the pyramids have triangular bases called as triangular pyramids, and so on. Triangular pyramid have three triangles meet at a common point, called apex, which come off of a common triangular base..

A triangular pyramid is a three -dimension figure with three faces that are triangles and base that is again a triangle.

## What is a Triangular Pyramid

A pyramid is described by the shape of its base. A pyramid having triangular base is called as a triangular pyramid. A triangular pyramid is a pyramid having a triangular base. The tetrahedron is a triangular pyramid having congruent equilateral triangles for each of its faces.

## Volume of a Triangular Pyramid

Triangular pyramid is a three-dimensional figure with four triangular bases. The volume of a pyramid in cubic units. The length of the triangular base is 'a' and 'H' is the height, distance from the base to the apex and 'h' be the slant height.

The volume of a triangular pyramid is:

V = $\frac{1}{3}$ A H

A = Area of the triangular base.

H = Distance from the base to the apex.(Height).

Here
Area of the triangular base (A) = $\frac{1}{2}$ah

=> Volume of Pyramid = $\frac{1}{3}$ * ($\frac{1}{2}$ah) * H

= $\frac{1}{6}$ahH.

Formula:

V = $\frac{1}{6}$ahH

Where,
V = Volume of the triangular pyramid.
a = Side of the triangular base.
h = Slant height or Apothem length.
H = Height of the pyramid, distance from the base to the apex.

Example:

Find the volume of the triangular pyramid with the side 12 cm, slant height 10 cm and height of the pyramid is 17 cm.

Solution:

Step 1:
Side of the triangular base = 12 cm
Slant height of the triangular pyramid = 10 cm
Height of the pyramid (H) = 17 cm

Step 2:

Area of the triangular base = $\frac{1}{2}$base * height

=> A = $\frac{1}{2}$ * 12 * 10

= 60

=> A = 60 cm2

Step 3:

Volume of the triangular pyramid (V) = $\frac{1}{3}$AH

=> V = $\frac{1}{3}$ * 30 * 17

= 170 cm3 .

Hence the volume of the triangular pyramid is 170 cm3 .answer

## Surface Area of a Triangular Pyramid

A Triangular Pyramid is a pyramid having a triangular base. The area of the surfaces of the object is called its surface area. The surface area of a pyramid in square units. The triangular pyramid is one third of a triangular prism of equal base and altitude. The length of the triangular base is 'a' and 'H' is the height, distance from the base to the apex, 'h' be the apothem length and '$l$' is the slant height.

Triangular Pyramid Formula :

Area of Base(A) = $\frac{1}{2}$ * a * h

Surface Area of Triangular Pyramid = ($\frac{1}{2}$ * a * h) + ($\frac{3}{2}$a$l$) = A + ($\frac{3}{2}$a$l$)

Formula:

SA = A +
$\frac{3}{2}$a$l$

Where,
SA = Surface area of triangular pyramid
A = Area of base
a = Side of the triangular base.
$l$ = Slant height.

Example:

Find the surface area of a triangular pyramid with the given apothem length 3, side 4, height 5 and the slant height 6.

Solution:

Step 1:

Apothem length (h) = 3
Side of base (a) = 4
Height of pyramid (H) = 5
Slant height ($l$) = 6

Step 2:

Find the area of the base.

Area of the base(A) = $\frac{1}{2}$ * a * h = $\frac{1}{2}$ * 4 * 3 = 6

=> A = 6

Step 3:

Find the surface area of pyramid.

Surface Area of Pyramid = A + $\frac{3}{2}$a$l$

SA = 6 + $\frac{3}{2}$ * 4 * 6

= 6 + 36

= 42

Hence surface area of the triangular pyramid is 42 units. answer