When you look around the room in your house, can you see anything that has a shape of a rectangular box? Maybe furniture, a book or the TV. This shape is a cuboid. Ice-cubes, dice, Rubik cube are all examples of a cuboid. Let us find the surface area and volume of cuboid and cubes.
What is a Cube?
The cube is a three-dimensional structure which is formed when six identical squares bind to each other in an enclosed form. A cube generally has 6 faces, 12 edges, and 8 vertices.
When a number is multiplied by itself three times, then the formed is a cube number. For example, 3 × 3 ×3 = 27 is a cube number.
Volume of Cube
The volume of a cube is length (l) × breadth (b) × height (h), since l = b = h in the cube, its sides can be represented as l = b = h = a. Therefore,
Volume of Cube = a³
where a is the measurement of each side of the cube. Hence, the volume of the cube of side 1 cm will be equal to 1 cm × 1 cm × 1 cm = 1 cm³. One should also be handy about the properties of cube numbers, like:
- 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 Natural Numbers
2³ = 8
3³ = 27
4³ = 64
5³ = 125
6³ = 216
7³ = 343
8³ = 512
9³ = 729
10³ = 1000
Numerical value obtained after cubing any given number is called a Perfect Cube. Cubes of even numbers are always even. Cubes of odd numbers are always odd.
What is a Cuboid?
A solid all of whose sides are rectangles are cuboids. Cuboid is a 3-dimensional shape or a polyhedron, having 6 rectangular sides called faces. It has 8 vertices and 12 edges. Each face is a rectangle and all the cuboids corner are vertices are 90-degree angles.
How to identify a cuboid?
In a cuboid, each face is a rectangle and the corners or the vertices are 90-degree angles. Moreover, the opposite faces are always equal. The simplest example is a book. It has 6 surfaces of which each opposite pair is of the same dimensions. Let l be the length, b be the breadth and h be the height of the cuboid.
Area of 4 side faces = 2 ( l × b ) + 2 ( b× h ) = 2 ( l × b ) × h = perimeter of base ×height
Area of 4 side faces is also called a lateral surface area. For example, the area of 4 walls of a room. Therefore,
Total surface area of a cuboid = 2 ( l × b ) + ( l× h ) + ( b× h )
Volume of Cuboid
Volume of cuboid = length × breadth × height
Or, volume of cuboid (v) = l × b × h
Also, the length of the longest diagonal of a cuboid = √ (l² + b² + h²)
Solved Examples For You
Q1. If six cubes of 10 cm edge are joined end to end, then the surface area (in sq. cm) of the resulting solid is
Solution: C. If six cubes are joined together, the new length of the cuboid formed = 60cm, breadth 10 cm and height = 10 cm. Now the surface area of cuboid formed
= 2 ( l × b ) + ( b× h ) + ( l× h )
i.e, = 2 ( 60× 10 ) + ( 10× 10 ) + ( 60× 10)
= 2 ( 1300)
= 2600 cm²
- 20 cm
- 10 cm
- 30 cm
- 40 cm
Solution: B. Volume of cuboid = length × breadth × height = Base area × height
Base area = 2500/25 = 100 cm²
Length = √100 = 10 cm
Now, 1000 = 10 ×10 = new height
New height = 1000/ 10 × 10 = 10 cm
Yes, the new block was cubical.