# Chapter 6 – Mensuration

The following Topics and Sub-Topics are covered in this chapter and are available on MSVgo:

Introduction

Mensuration means measuring geometric figures and their parameters like length, volume, shape, surface area, lateral surface area, etc. While studying a closed plane figure, you need to know its perimeter and area. The perimeter is the distance around its boundary, and its area is the region covered by it. For example, before fencing a garden, you must know the perimeter to check the length of the fence required. While, to know about the amount of land occupied by the garden, you must know the area of the garden.

We can find the area and perimeter of various plane figures such as triangles, rectangles, circles etc. and solid surfaces like a cube, cuboid, sphere, etc.

#### Area and Perimeter of a Triangle

Draw an equilateral triangle of each side measuring 6 cm and named as ABC. If you had to stick colourful tape on the triangle’s boundary, how much longer will you require? To find this, you must know the perimeter of the triangle.

The perimeter of Triangle = Sum of all the sides (Isosceles and Scalene Triangle)

Or

The perimeter of Triangle = 4 x side (Equilateral triangle)

For the above triangle having sides of 6 cm,

Perimeter of triangle ABC = 4 x 6

Perimeter = 24 cm

It means you will need 24 cm long tape.

Now, if you want to colour the triangle, you must know the area of the triangle.

Area of Triangle = ½ × b × h

Where b= Base of the triangle and h = height of the triangle

If for this triangle, height was 4 cm, then

Area of Triangle ABC = ½ × 6 × 4

Area of Triangle ABC = 12 cm

#### Area and Circumference of a Circle

If you are moving in a circular path and want to know the distance covered by you to complete one round, you must find the circle’s perimeter.

Circumference of the circle = 2 π r

Where the value of π is 22/7 or 3.14 and r is the radius of the circle.

If I have a circle with a radius of 14 m, then

Circumference of the circle = 2 x 22/7 x 14

Circumference of the circle = 88 m

If you were given two medium-sized pizzas and one large size pizza and you were asked to compare which one has more quantity, how will you find it? You need to find the medium pizza area and the large pizza to compare which one is larger.

Area of circle = π r2

If the medium pizza radius was 14 cm and the large pizza was 28 cm, then.

Area of one medium pizza = 22/7 x (14)2 = 616 cm2

Area of two medium pizza = 2 x 616 = 1,232 cm2

Area of one large pizza = 22/7 x (28)2 = 2,464 cm2

This signifies that one large pizza is larger than two medium pizzas.

#### Solids Surface - Area and Volume of Cube and Cuboids

Solid surfaces or objects are those which have three dimensions. All six faces are squares and identical in a cube-like Rubik’s cube. All six faces are rectangular, and the opposite faces are identical. If there are three pairs of identical faces, such an object is called a cuboid, like a duster used in your class.

Area and Volume of Cube

If you have asked the painter to paint a cubical room’s walls and tell you the cost per unit area, how will you find how much money you need to pay him? You need to find the lateral surface area of the cubical room first.

Lateral surface area of cube = 4 a2

a = side

If side of cube measures 250 cm, then

Lateral surface area of room = 4 x (250)2

Lateral surface area of room = 2,50,000 cm2

If you need to paint a cubical box of side 8 cm, you need to consider its total surface area.

The total surface area of cube = 6 a2

So, Total surface area of cubical box = 6 x (8)2 = 384 cm2

To find the amount of water that can be filled inside the cubical container, you must find the volume of a cube,

The volume of cube = a3

For the above mentioned cube, volume will be = (8)3 = 512 cm3

#### Area and Volume of Cuboid

If you have a cuboidal-shaped geometry box and the teacher asked you to colour the box, leaving its top and bottom, how much area will you colour? To find this, you need to find the lateral surface area of your cuboidal box.

Lateral Surface area of cuboid = 2h(l + b)

Where, h = height; l = length; b = base

If you have a cuboidal box and you need to cover it entirely with glitter paper, then how much glitter paper will you require? You can find that by calculating the total surface area of the cuboid.

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

If you have a cuboidal mould to set the chocolate, how much liquid chocolate do you need to fill in it? You can know that with the help of finding the volume of the cuboidal box.

Volume of cuboid = l × b × h

#### Conclusion

Mensuration is studying different parameters in the plane figures and solid surfaces with the area, perimeter, circumference and volume. For a polygon like a triangle, the area and perimeter of different triangle types can be found differently. For solid surfaces like cube and cuboid, you need to understand the question first to determine whether you need to find the lateral surface area or total surface area.

#### FAQs

Q1. What is mensuration?

A1. It is the measurement of geometric figures and their parameters like length, volume, shape, surface area, lateral surface area, etc.

Q2. What do you mean by the surface area of the object?

A2. It is the total area covered by the surface of the solid or 3-dimensional object.

Q3. Is surface area similar to volume?

A3. No, the surface area is a two-dimensional quantity, while volume is a three-dimensional quantity.

Q4.  What is the use of mensuration in daily life?

A4. Uses of mensuration in daily life are:

1. Amount of carpet required in a room
2. Fencing for the garden
3. Reading the roadmap
4. The volume of soil needed to fill a ditch

Q5. What is the formula of lateral surface area for cube and cuboid?

A5. Lateral surface area of cube = 4 a2

Lateral Surface area of cuboid = 2h(l + b)

There is a lot that you have learned about the area, perimeter, and volume of the triangle, circle, cube, and cuboid. To know about plant life further, you can download the MSVgo app, which offers a large base of learning videos, quizzes, interactive games, and many more to the students till grade 12.

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