Mensuration

Cube and Cuboid

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.

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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.

Cuboid

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

What is Area of Rectangle?

Cubes of Natural Numbers

1³= 1
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?

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.

What is the Area of Square?

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)

What is Area of Trapezium, Parallelogram, and Rhombus?

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

  1. 3600
  2. 3000
  3. 2600
  4. 2400

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

Q2. 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.
  1. 20 cm
  2. 10 cm
  3. 30 cm
  4. 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.

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