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# Surface Area of a Rectangular Prism

There are many geometrical and non geometrical objects that we see in our surroundings every day. Usually, some are two dimensional, while others are defined in three dimensions. A prism is a very commonly used three dimensional object. A prism is composed of two base surface facing each other. These faces are joined together with help of lateral surfaces. A prism is usually made up of a transparent or translucent material. Few examples of prisms are triangular prism, rectangular prism, pentagonal prism, hexagonal prism etc.

A rectangular prism is a prism that has two rectangles as the base surfaces and four lateral rectangle surfaces joining them. The rectangular prism is a shape with two rectangles as bases and connected together by four faces. prism is also another name of a cuboid. Given below is a picture of rectangular prism of height 'H', length 'L' and breadth 'B'.

Surface Area of a Rectangular Prism is the measure of how much exposed area a prism has. Since rectangular prism is a cuboid, the surface area of cuboid and surface area of rectangular prism are same. Surface Area is expressed in square units. Let us learn more about the surface area of rectangular prisms.

 Related Calculators Surface Area of a Rectangular Prism Calculator Prism Surface Area Calculator Surface Area of a Rectangular Pyramid Calculator Surface Area of a Triangular Prism Calculator

## Surface Area of a Rectangular Prism Formula

As explained earlier, the rectangular prism has two rectangular bases connected together by rectangular faces. So, the surface area will be the sum of the area of the two bases and the area of the four faces. For a rectangular prism, the bases will have length L and breadth B and the height or the distance between the bases is H.

The surface area of the two bases = 2 $\times$ L $\times$ B

The surface area of the four sides of the prism = 2 $\times$ L $\times$ H + 2 $\times$ B $\times$ H = 2H (L+ B)

The surface area of the prism = 2LB + 2H (L+ B)

Formula for total surface area of a rectangular prism = 2LB + 2H (L+ B)

where, L - Length of rectangular prism
B - Breadth of rectangular prism
H - Height of rectangular prism

## Lateral Surface Area of a Rectangular Prism

The lateral surface area of a rectangular prism is the area can be worked out by adding the areas of each side together. The lateral surface area of a rectangular prism is equal to the perimeter of the base times the height of the prism.

=> Lateral surface = Ph

Where, P is the perimeter of a base and h be the height of the prism.

Perimeter of the rectangular prism is,

P = 2(Length + Width)

Lateral surface area of rectangular prism = PH = 2(Length + Width) $\times$ Height of Prism

## Surface Area of a Right Rectangular prism

A right prism is a 3D solid with two parallel, congruent polygons for bases and rectangles for lateral faces.

Surface area of the prism = Area of two bases + Lateral area of prism

Total surface area of right rectangular prism, S = 2B + Ph

where, B = Length * Width (Area of a base)
P = 2(Length + Width) (Perimeter of a base) and
h = Height of prism

The surface area of a right rectangular prism is 856 sq. cm. The height is 14 cm. The length of the base is 2 cm more than the width of the base. Let us find the width of the prism.

Let x be the width of the prism, therefore x + 2 be the length of the prism.

S
urface area of a right rectangular prism = 856 sq. cm (Given)

We know that, S = 2B + Ph

Here, B =
(x + 2)x and P = 2(x + 2 + x) and h = 14

According to statement:

856 =
2(x + 2)x + 2(x + 2 + x) 14

428 = (x$^2$ + 2x) + (2x + 2) 14

428 =
x$^2$ + 30x + 28

or x$^2$ + 30 x
- 400 = 0

By solving this equation, we have

x = 10, -40

Length can not be negative, so neglect x = -40

=> x = 10 (width of the prism)

Hence, the width of the right rectangular prism is 10 cm.

## Find the Surface Area of a Rectangular Prism

Given below are some of the examples to find the surface area of a rectangular prism.

### Solved Examples

Question 1: Calculate the surface area of the rectangular prism with length = 4 cm, breadth = 3 cm and height of prism = 8 cm.
Solution:

Dimensions of the rectangular prism are:

L = 4 cm, B = 3 cm and H = 8 cm (given)

The surface area of the prism = 2LB + 2H (L+ B)

= 2 $\times$ 4 $\times$ 3 + 2 $\times$ 8 $\times$ (4 + 3)

= 24 + 16 $\times$ (7)

= 24 + 112

= 136

Therefore, the surface area of the rectangular prism is 136 cm2.

Question 2: Calculate the surface area of the rectangular prism with length = 5 cm, breadth = 2.5 cm and height of prism = 9 cm.
Solution:

Given, L= 5 cm, B = 2.5 cm and H = 9 cm

The surface area of the prism = 2LB + 2H (L+ B)

= 2 $\times$ 5 $\times$ 2.5 + 2 $\times$ 9 $\times$ (5 + 2.5)

= 25 + 18 $\times$ (7.5)

= 25 + 135

= 160

The surface area of the rectangular prism is 160 cm2.

Question 3: Find the surface area of a box in the shape of a right prism whose bases are rectangles with side lengths 6 cm and 4 cm and whose height is 10 cm.
Solution:

Given, L = 6 cm, B = 4 cm, H = 10 cm

The surface area of the prism = 2LB + 2H (L+ B)

= 2 $\times$ 6 $\times$ 4 + 2 $\times$ 10 $\times$ (6 + 4)

= 48 + 20 $\times$ (10)

= 48 + 200

= 248

The surface area of the box is 248 cm2.