Pyramid is a solid having any plane figure for its base and triangles for its sides. A pyramid is described by the shape of its base. The number of triangular faces depends on the number of sides of the base. For instance, a pyramid with a rectangular base has four triangular faces, whereas a pyramid with a hexagonal face is made up of six triangular faces, and so on.
A rectangular pyramid is a three -dimension figure with four faces that are triangles and base that is a rectangle.

A pyramid is a solid figure with a polygonal base and triangular faces that meet at a common point. A pyramid is described by the shape of its base. A pyramid having rectangular base is called as a rectangular pyramid. Rectangular pyramid have four triangles meet at a common point, called apex, which come off of a common rectangular base.

Rectangular Pyramid

Where,
l, b, be the length and width of the rectangular base of the rectangular pyramid.

$s_1$ and $s_2$ be the slant lengths of the triangles with base length and width respectively.

h is the height of the pyramid.

A rectangular pyramid is built on a rectangular base, with two sets of triangles forming the sides. Each set of triangles will be equal in side length and height, but will have different base lengths. The volume of a pyramid is one third of the product of its altitude and the area of its base. The volume of a pyramid in cubic units. The length and width of the rectangular base are 'a', 'b' and h is the height, distance from the base to the apex.

The volume, V, of a rectangular pyramid is,

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

Where, A = length * width = ab (
Area of the base)

=> V =
$\frac{1}{3}$abh

Formula:

V = $\frac{1}{3}$abh cubic units

where, V is the volume of the pyramid.

'a' and 'b' be the sides of the rectangular base
.

h is the height, distance from the base to the apex.


Example:

Find the volume of a rectangular based pyramid whose base is 5 x 6 cm and height is 7 cm.

Solution:
Step 1:

Given
Length of base of the rectangular pyramid = 5 cm
Breadth
of base of the rectangular pyramid = 6 cm
Height
of base of the rectangular pyramid = 7 cm

Step 2:

Find the area of the rectangle:

A = Length * Breadth


= 5 * 6

= 30

=> A = 30 cm2 .

Step 3:

Volume of the rectangular pyramid (V) = $\frac{1}{3} $Ah.

=> V =
$\frac{1}{3}$ * 30 * 7

= 70

=>
Volume of the rectangular pyramid is 70 cm3 .
The surface area of a regular square pyramid is the sum of area of the base and area of four isosceles triangles. But for rectangular pyramid, with rectangular base, we can not use the standard formula of surface area because there is more than one slant height ($s_1$ and $s_2$). A rectangular pyramid is made up of one rectangular base and four triangles going up from the base to the top of the pyramid. The surface area is the area of all five parts added together. The area of the rectangular base with length (l) and width (b) is 'lb'.

Formula:

Surface area of Rectangular pyramid (SA) = lb + l $s_1$ + b $s_2$.

where,
'l' is length, 'b' is width.

$s_1$ is the slant length of the triangles with base 'l'.

$s_2$ is the slant length of the triangles with base 'b'.


Example:

Find the surface area of a rectangular based pyramid whose base is 12x8 m and slant heights are 10m and 12 m.

Solution:
Pyramid

Step 1:

Given
Length of base of the rectangular pyramid (l) = 12m
Breadth
of base of the rectangular pyramid (b)= 8m
Slant height
of the triangle with base length (
$s_1$) = 10m
Slant height of the triangle with base width ($s_2$) = 12m

Step 2:

Find the area of the rectangle:

A = Length * breadth


=> A = 12 * 8

=> A = 96 m2 .

Step 3:

Surface area of rectangular pyramid (SA)

SA = A + l $s_1$ + b $s_2$.

=> SA = 96 + 12 * 10 + 8 * 12

= 96 + 120 + 96

= 312

Hence the surface area of rectangular pyramid is 312 m2 .