When you look around the room in your house, you can see many objects that have the shape of a rectangular box. For example, furniture, books and TV are in the shape of a rectangular box. This shape is a cuboid. On the other hand, objects such as ice, dice and Rubik’s cube are examples of another 3D shape called a cube. As a matter of fact, a cube is a special type of cuboid in which all sides are squares and identical. Let us learn more about cubes and cuboids.

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## What is Cuboid?

In geometry, a cuboid is a three-dimensional shape in which all sides are rectangles. It is a polyhedron, having 6 rectangular sides called faces, 8 vertices and 12 edges. These rectangular faces are at right angles to one another. Thus, all angles in a cuboid are right angles.

**How to identify a cuboid?**

In a cuboid, each face is a rectangle and the corners or the vertices are 90-degree angles. Also, the opposite faces are always equal. For example, a book is a cuboid. It has 6 surfaces of which each opposite pair is of the same dimensions.

**Download Mensuration Cheat Sheet PDF Below**

**Total Surface Area of Cuboid**

If l is the length, b is the breadth and h is the height of the cuboid, then the sum of areas of six rectangles of a cuboid gives the total surface area of the cuboid. The formula for it is given below.

**Total surface area of a cuboid = 2 [( l × b ) + ( l× h ) + ( b× h )]**

**Browse more Topics under Mensuration**

- Cylinder
- Circles
- Polygons
- Rectangles and Squares
- Trapezium, Parallelogram and Rhombus
- Area and Perimeter

**Lateral Surface Area of Cuboid**

The sum of the area of 4 side faces i.e. leaving the bottom and the top face gives the lateral surface area of a cuboid. An example of the lateral surface area is the sum of the area of the four walls of a room. The formula to calculate the lateral surface area of a cuboid is

Area of four sides = 2 ( l × h ) + 2 ( b× h ) = 2 ( l + b ) × h = perimeter of base ×height

or simply,

**Lateral surface area of a cuboid = 2(l+b)h**

**Volume of Cuboid**

The volume of a cuboid can be found by multiplying the base area with the height. Therefore,

volume (V) = A x h = (l x b) x h. In simple terms,

**Volume of cuboid (v) = l × b × h**

where l is the length, b is the base and h is the height of the cuboid.

**Diagonal of a Cuboid**

The length of the longest diagonal of a cuboid is given by

**Length of diagonal of cuboid = √ (l² + b² + h²)**

What is Area of Trapezium, Parallelogram, and Rhombus?

## What is a Cube?

A cube is a three-dimensional object which is formed when six identical squares bind to each other in an enclosed form. A cube has 6 faces, 12 edges, and 8 vertices. In other words, a cuboid whose length, breadth and height are equal is called a cube.

**Volume of Cube**

The formula to calculate the volume is length (l) × breadth (b) × height (h). Since l, b, h in a cube measure the same, its sides can be represented with a. So, l = b = h = a. Therefore,

volume of cube = l x b x h = a x a x a. Or simply,

**Volume of Cube = a³**

where a is the measurement of each side of the cube.

For example, the volume of the cube of side 1 cm will be equal to 1 cm × 1 cm × 1 cm = 1 cm³.

When a number is multiplied by itself three times, then the resulting number is called cube number. For example, 3 × 3 ×3 = 27. 27 is a cube number. Below is a list of cubes of first ten natural numbers. This list will come in handy for quick calculations.

**Cubes of First 10 Natural Numbers**

1³= 1

2³ = 8

3³ = 27

4³ = 64

5³ = 125

6³ = 216

7³ = 343

8³ = 512

9³ = 729

10³ = 1000

The numerical value obtained after cubing any given number is called a **Perfect Cube**. Knowing the properties of cube numbers will be helpful in calculating the volume of a cube.

**Properties of Cube Number**

- Cubes of positive numbers are always positive. For example, cube of +4 is = (+4) × (+4) × (+4) = +64
- Cubes of negative numbers are always negative. For example, cube of -4 is = (-4) × (-4) × (-4) = -64
- Cubes of even numbers are always even.
- Cubes of odd numbers are always odd.

## Solved Examples For You

**Multiple Choice Questions**

**Question 1. **If six cubes of 10 cm edge are joined end to end, then the surface area (in sq. cm) of the resulting solid is

- 3600
- 3000
- 2600
- 2400

**Answer:** C. If six cubes are joined together, then the resulting solid is a cuboid. The dimensions of the solid are as follows,

length =60cm, breadth = 10 cm and height = 10 cm. Now, the total surface area of a cuboid is given by

2 [( l × b ) + ( b× h ) + ( l× h )]

By substituting the values, we get,

= 2 [( 60× 10 ) + ( 10× 10 ) + ( 60× 10)] = 2 ( 1300) = 2600 cm²

**Question 2.**John built a rectangular cardboard box 25 cm high with a square base and a volume of 2500 cm³. Then he realized he did not need a box that large, so he chopped off the height of the box reducing its volume to 1,000 cm³. Was the new box cubical? Also, state its’ height.

- 20 cm
- 10 cm
- 30 cm
- 40 cm

**Answer:** B. Volume of cuboid (V) = length × breadth × height = Base area × height.

Given, V = 2500 cm^{3}, height = 25 cm. By substituting the values in the formula, we get

Base area = 2500/25 = 100 cm²

Given that the base is a square. This means that the length = breadth. Therefore, the length of square base = √*100 = 10 cm
*After chopping, new volume = 1000 = 10 ×10 x new height

Therefore, new height = 1000/ 10 × 10 = 10 cm

As all the dimensions of the solid, l, b, h measure the same, the resulting solid is a cube.

**Answer The Following**

**Question 3: **Find the surface area of a cube whose edge is 5 cm?

**Answer:** Cube is a type of cuboid in which the length, breadth and height measure the same. The formula to calculate the total surface area of a cuboid is 2 [( l × b ) + ( l× h ) + ( b× h )]. As l=b=h=a, the total surface area of cube= 2 [( a × a ) + ( a× a ) + ( a× a )] = 2 [3a^{2}] = 6a^{2}. To sum up,

**Total surface area of cube = 6a ^{2}**

Given that a = 5 cm. Thus,

T.S.A =6(5^{2}) = 6 X 25 = 150 cm^{2}

**Question 4: **Define volume.

**Answer:** Volume is the amount of space that is occupied by a three-dimensional solid object. The unit of measurement of volume is the cubic meter.

**Question 5: **What is the lateral surface area of a cube of edge 10cm?

**Answer:** We know that the lateral surface area of a cuboid is given by** **2(l+b)h. As a cube is a cuboid in which l=b=h=a, lateral surface area of cube = 2(a+a)a = 4a^{2}. In simple terms

**Lateral surface area of cube = 4a ^{2}**

Given that a = 10 cm. Thus,

LSA = 4(10^{2}) = 400 cm^{2}

**Question 6: **How do you measure the volume of water?

**Answer: **Basically, we can’t measure the volume of water until it is stored in a container, which can be a cube, cuboid, cylinder, cone, etc. And once it is inside a container we have to calculate the volume of the container to know the volume of water.

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