# Volume and Surface area of cuboid

In this chapter we will learn the basic concept of cuboid along with the formula to calculate volume, total and lateral surface area.

At the end of the chapter, some problems are also given for practice.

## What is cuboid ?

The cuboid is a three dimensional figure formed by joining 6 rectangular planes at right angle to each other.

The dimension of cuboid is expressed through length, breadth and height. The size of the cuboid is determined by these dimension.

Properties of cuboid are;

(a) A cuboid has 6 faces, 8 vertices and 12 edges

(b) The opposite faces of the cuboid are parallel to each other.

(c) Face diagonals
Diagonal drawn by joining opposite side of the rectangular face is called face diagonals.

Cuboid is made of 6 rectangular faces and each face contain two diagonals. So in a cuboid, there are total of 12 rectangular diagonals.

(d) Space diagonals

Diagonal formed by joining opposite edge of cuboid is called space diagonals.

In a cuboid, there are 4 space diagonals.

### Volume of cuboid

The total space enclosed by the cuboid is called volume of cuboid.

For better understanding, consider an open box in the form of cuboid. Now fill the box with water to the brim. The total volume of water in the cuboid is equal to the volume of cuboid.

If “l”, “b” and “h” are the length, breadth and height of given cuboid, then the formula for volume of cuboid is given as;

Volume of cuboid = Length x Breadth x Height

Volume of cuboid = L x B x H

We get volume data in cubic units ( \mathtt{cm^{3} \ or\ m^{3} \ etc..} )

### Total surface area of cuboid

The area of all the walls covered by the cuboid is called total surface area of cuboid.

We know that cuboid is made of 6 rectangular faces. The detail of each face is given below;

1st Face ⟹ Length x Breadth
2nd Face ⟹ Breadth x Height
3rd Face ⟹ Length x Height
4th Face ⟹ Length x Breadth
5th Face ⟹ Breadth x Height
6th Face ⟹ Length x Height

The formula for total surface area is given as;

Total Surface Area = 2 (lb + bh + hl)

We get the total surface area in square units ( \mathtt{cm^{2} \ or\ m^{2} \ etc..} ) .

### Lateral surface area of cuboid

The total area of the walls of cuboid (all faces excluding top & bottom) is called lateral surface area of cuboid.

The formula for lateral surface area is given as;

Lateral Surface area = 2 (lh + bh)

We get the lateral surface area in square units ( \mathtt{cm^{2} \ or\ m^{2} \ etc..} ).

I hope you understood the basic concept. Let us solve some problems related to the concept.

### Volume and surface area of cuboid – Solved problems

Question 01
Given is the cuboid of length 20 cm, breadth 15 cm & height 10 cm. Find the volume, total and lateral surface area of cuboid.

Solution
Volume = Length x Breadth x Height

\mathtt{Volume\ =\ 20\times 15\times 10\ cm^{3}}\\\ \\ \mathtt{Volume\ =\ 3000\ cm^{3}}

Hence, the volume of given cuboid is 3000 cu. cm.

Total Surface area = 2 (lb + bh + hl)

\mathtt{Total\ surface\ area=2\ ( 20.15+15.10+10.20)}\\\ \\ \mathtt{Total\ surface\ area=2\ ( 300+150+200) \ =\ 1300\ cm^{2}}

Hence, total area of cuboid is 1300 sq. cm

Lateral Surface area = 2 (lh + bh)

\mathtt{Lateral\ surface\ area=\ 2\ ( 20.10\ +\ 15.10)}\\\ \\ \mathtt{Lateral\ surface\ area\ =\ 2\ ( 200+150) =\ 700\ cm^{2}}

Hence, the lateral surface area is 700 sq. cm

Question 02
Stacey created a cuboid box to keep her monthly savings. The dimension of the box is length 30 cm, breadth 15 cm and height 30 cm. She want to cover whole box with colored paper. If area of a colored paper is 40 sq. cm, find the total number of decorated paper needed to cover the whole box.

Solution
First calculate the total surface area of the cuboid.

Total surface area = 2 (lb + bh + hl )

\mathtt{Total\ surface\ area=2\ ( 30.15+15.30+30.30)}\\\ \\ \mathtt{Total\ surface\ area=2\ ( 450+450+900) \ =\ 3600\ cm^{2}}

Hence, 3600 sq. cm is the total surface area of given box.

Now let’s calculate the total piece of decorated paper use to cover the whole box.

Paper used = Total Area / Area of one paper

Paper used = 3600 / 40 = 90 peice

Hence, total of 90 piece of paper is required to cover the box.

Question 03
John decide to paint side walls of the room of length 3 meter, breadth 4 meter and height 8 meter. If the cost of paint is 3$per sq. meter. Find the total cost to complete the work. Solution Calculate the lateral surface area of the cuboid. Lateral Surface area = 2 (lh + bh) Lateral surface area = 2 (3 x 8 + 4 x 8) = 112 sq. meter Now calculate the cost to paint all four walls. Total cost = 112 x 3 = 336$

Hence, 336\$ is the total expenditure.

Question 04
The length and breadth of hall is 18 meter and 12 meter respectively. If the sum of areas of side walls is equal to sum of floor and roof. Calculate the height of cuboid room.

Solution
Let the height of room be ” h “.

Area of floor = Length x Breadth = 18 x 12 = 216 sq. m

In cuboid, area of floor = area of roof.

Total area of floor & roof = 216 + 216 = 432 sq. m

Area of four side walls = Lateral surface area

Lateral surface area = 2 (lh + bh)

Lateral surface area =2 (18h + 12h) = 60h

As per the question;

60 h = 432

h = 432 / 60 = 7.2 meter

Hence, height of the given room is 7.2 meter

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