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Mensuration

Cube and Cuboid

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?

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.

What is Area of Rectangle?

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.

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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 )]

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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.

Cube

What is the Area of Square?

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

  1. 3600
  2. 3000
  3. 2600
  4. 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.
  1. 20 cm
  2. 10 cm
  3. 30 cm
  4. 40 cm

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

Given, V = 2500 cm3, 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 [3a2] = 6a2. To sum up,

Total surface area of cube = 6a2

Given that a = 5 cm. Thus,

T.S.A =6(52) = 6 X 25 = 150 cm2

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 = 4a2. In simple terms

Lateral surface area of cube = 4a2

Given that a = 10 cm. Thus,

LSA = 4(102) = 400 cm2

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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