CAT 2025 Lesson : Mensuration - 2-D Figures

1. Introduction

Mensuration, meaning measurement, is a branch of mathematics which deals with the calculation of measurements such as distance, area, volume, etc. In CAT and other management entrance tests, we are tested on 2-dimensional (2-D) and 3-dimensional (3-D) figures only, which are detailed below.

Figure	2-D Figures	3-D Figures
Definition	A shape that can be drawn on a flat piece of paper or any other flat plane.	A solid object which occupies space and has volume.
Dimensions	Length and Breadth	Length, Breadth and Height
Example	Triangle, Quadrilateral, Pentagon, etc.	Cube, Cylinder, Sphere, etc.,

Mensuration pertaining to 2-D figures has been covered in the previous lessons in geometry, i.e., Lines & Triangles, Quadrilaterals and Circles. This lesson would serve as a recap for mensuration pertaining to 2-D figures and will provide mensuration for 3-D figures in detail.

1.1 Basic Conversions of units

Knowing these conversions will be helpful while solving mensuration questions.

Length / Distance	Area	Volume
1 Kilometre (km) = 1000 m		1 Kilolitre (kl) = 1000 l
1 Hectometre (hm) = 100 m	1 Hectare (ha) = 100 a	1 Hectolitre (hl) = 100 l
1 Decametre (dam) = 10 m	1 Decare (daa) = 10 a	1 Decalitre (dal) = 10 l
1 Metre (m)	1 Are (a) = 100 $\bold{m^2}$	1 Litre (l) = 1000 $\bold{cm^3}$
1 Decimetre (dm) = 0.1 m	1 Deciare (da) = 0.1 a	1 Decilitre (dl) = 0.1 l
1 Centimetre (cm) = 0.01 m	1 Centiare (ca) = 0.01 a	1 Centilitre (cl) = 0.01 l
1 Millimetre (mm) = 0.001 m		1 Millilitre (ml) = 0.001 l

Note: Please note the are and litre equivalents provided in the table above.

When area or volume is provided as a unit of length (say metres), it can be converted to a different unit by squaring or cubing their respective length conversions. The following table contains a few examples.

Area	Volume
$13 \ m^{2} = 13 \times (100 \ cm)^2 = 130,000 \ cm^2$	$5 \ m^{3} = 5 \times (100 \ cm)^3 = 5,000,000 \ cm^3$
$24 \ km^{2} = 24 \times (1000 \ m)^2 = 24,000,000 \ m^2$	$8 \ km^{3} = 8 \times (1000 \ m)^3 = 8,000,000,000 \ m^3$
$5,000 \ mm^{2} = 5,000 \times (0.1 \ cm)^2 = 50 \ cm^2$	$7,000 \ mm^{3} = 7,000 \times (0.1 \ cm)^3 = 7 \ cm^3$

Inches and feet are other measures of length wherein 1 inch = 2.54 cm and 1 feet = 12 inches.

With feet being a measure of length, areas are also mentioned with square feet (sqft) as a unit.

Acre is another unit used for area, wherein 1 acre = 43,560 sqft

2. 2-D Figures

A shape with only two dimensions (length and breadth) and no thickness is a 2D figure. We can calculate only the perimeter and area for these figures. Some examples of 2-D figures are square, rectangle, triangle, etc. Each of the figures listed below have been covered in the three preceding lessons under Geometry.

Shape	Perimeter	Area	Figure
Scalene Triangle	$a + b + c$	$\dfrac{1}{2} \times b \times h = \dfrac{1}{2} \times ab \times sin\theta$ = $\sqrt {s(s - a) (s - b) (s - c)}$ = $\dfrac{abc}{4R} = s \times r$ (where $s = \dfrac{a+ b + c}{2}$, $R$ is the circumradius and $r$ is the inradius )
Isosceles Triangle	$2a + b$	$\dfrac{b}{4} \times \sqrt {4 a^2 - b^2}$
Equilateral Triangle	$3a$	$\dfrac{\sqrt3}{4} a^2$
Quadrilateral	AB + BC + CD + DA	$\dfrac{1}{2} \times$ AC $\times (h_1 + h_2)$
Parallelogram	$2(a + b)$	$b \times h$
Rhombus	$4a$	$\dfrac{1}{2} \times d_1 \times d_2$

Shape	Perimeter	Area	Figure
Rectangle	$2(l + b)$	$lb$
Square	$4a$	$a^2$
Trapezium	AB + BC + CD + AD	$\dfrac{1}{2} h (a + b)$
Circle	$2 \pi r$	$\pi r^2$
Semicircle	$\pi r + 2r$	$\dfrac{1}{2} \pi r^2$
Sector	$\left(\dfrac{\theta}{360 \degree} \times 2 \pi r \right) + 2r$	$\left(\dfrac{\theta}{360 \degree} \times \pi r^2\right)$