The knowledge of mensuration is very useful in our day to day life. Mensuration is being used by all us every day either directly or indirectly, for example - when we buy cloths of our shirts or pants, plot of land by area, milk/petrol by volume etc., we use mensuration.  In this lesson, we shall go ahead and learn mensuration, two dimensional shapes, three dimensional shapes and formulae used in mensuration.

## What is Mensuration?

In Latin, the word “mensuration” means “measurement” and hence we can define mensuration as the measurement of the geometric quantities, say length, breadth, area, perimeter etc. Mensuration is an important part of mathematics that studies about the measurements of geometrical objects such as – their area, volume and other parameters.

Broadly speaking, the mensuration is concerned of the process of measurement. The measurement is done using various geometric formulas and calculations. In mensuration, we make use of the data related to measurement like length, width, height, volume, area of a given shape or object. The results of using methods of mensuration are so accurate that they are considered to be better than the actual physical measurements.

## What is a 2D Shape?

The abbreviation of 2D is “two dimensional” which means a 2D shape is a geometrical figure that is enclosed by three or more than three straight lines or a closed in a circular curve. The 2D shapes exist in one plane. In other words, the shapes which have only two dimensions – known as length or breadth, they do not have height or depth, are known as 2D shapes.

The examples of few 2D shapes are circle, square, rectangle, triangle, parallelograms, rhombus, polygons etc. Some of the 2D shapes are shown below:

Image

Usually, area and perimeter are measured for 2D shapes.

## What is a 3D Shape?

The abbreviation of 3D is “three dimensional”. So, a 3D shape is defined as a geometrical shape which is covered by at least three planes/surfaces or circular face. The 3D shapes have three dimensions length, height (depth) and breadth, unlike 2D shapes. Another terminology for 3D shapes is “solids”. For example: sphere, hemisphere, cylinder, cone, cuboid, cube, pyramid, prism etc. Few images of 3D shapes are shown below:

image

Usually, surface area and volume are measured in 3D shapes.

## Mensuration Formulas

Formulas for 2D Shapes

Assume that $l, b$ represent length and breadth of a 2D shape.
(1) Rectangle

Perimeter = $2(l + b)$

Area = $l \times b$
(2) Square

Perimeter = $4 \times side$

Area = $(side)^2$
(3) Parallelogram

Perimeter = $2(l + b)$

Area = $l \times b$
(4) Triangle

Scalene Triangle

Let $a, b, c$ be the length of three sides of a triangle.

Perimeter $(2s)$ = $a + b + c$

Semi perimeter $(s)$ = $\frac{a+b+c}{2}$

Area = $\sqrt{s(s-a)(s-b)(s-c)}$  This is called Heron’s formula.

Area = $\frac{1}{2}$ $base \times height$

Isosceles Triangle

Let a being the measure of two equal sides and b be the base.

Perimeter = $2a + b$

Area = $\frac{b \sqrt{4a^2-b^2}}{4}$

Equilateral Triangle

Perimeter = $3 \times side$

Area = $\frac{\sqrt3}{4}$ $side^2$
(5) Trapezium

Perimeter = Sum of $4$ sides

Area = $\frac{1}{2}$ $\times$ sum of parallel sides $\times$ distance between parallel sides
(6) Rhombus

Perimeter = $4 \times side$

Area = $\frac{1}{2}$ product of diagonals
(7) Circle

Let $r$ be the radius of circle.

Diameter = $2r$

Circumference = $2 \pi r$

Area = $\pi r^2$
(8) Semicircle

Circumference = $\pi r$

Area = $\frac{1}{2}$ $\pi r^2$
Formulas for 3D Shapes

(Assume that $l, b, h$ stand for length, breadth, height respectively in a 3D shape.)
(1) Cuboid

Lateral surface area = $2(l + b) \times h$

Total surface area of cuboid = $2(lb + bh + hl)$

Volume = $l, b, h$
(2) Cube

Lateral surface area = $4a^2$

Total surface area = $6a^2$

Volume of cube = $side^3$
(3) Cylinder

Curved surface area = $2 \pi rh$

Total surface area = $2 \pi r(h + r)$

Volume = $\pi r^2 h$

Where, $r$ being the radius of base and $h$ is the height.
(4) Cone

Slant height $(l)$ = $\sqrt{h^2 + r^2}$

Curved surface area = $\pi r\ l$

Total surface area = $\pi r\ (l + r)$

Volume = $\frac{1}{3}$ $\pi r^2 h$
(5) Sphere

Curved surface area = Total surface area = $4 \pi r^2$

Volume = $\frac{4}{3}$ $\pi r^3$
(6) Hemisphere

Curved surface area = $2 \pi r^2$

Total surface area = $3 \pi r^2$

Volume = $\frac{2}{3}$ $\pi r^3$
(7) Prism

Lateral surface area = perimeter of base $\times$ height

Total surface area = Lateral surface area $+\ (2 \times$ area of base)

Volume = Area of base $\times$ height
(8) Pyramid

Lateral surface area = $\frac{1}{2}$ $\times$ Perimeter of base $\times$ slant height

Where, slant height = Height of the lateral triangular surface

Total surface area = Lateral surface area + area of base

Volume = $\frac{1}{3}$ $\times$ area of base $\times$ height