Chapter 5
Mensuration
Introduction
Mensuration is a field of mathematics that deals with studying geometric figures and measurements of their different parameters. It lays down the principles for estimating essential properties and equations related to various geometrical shapes and figures. There are mainly two types of shapes, namely 2D and 3D.
Types of Geometrical Figures
- 2D figures
Any figure that is enclosed by three or more straight lines or segments in a plane is called a 2D or plane figure. These shapes have only two kinds of dimensions, i.e., length and breadth. Common examples of 2D shapes are triangle, circle, and polygon. - 3D figures
Any figure that is surrounded by a number of planes or surfaces is called a 3D or solid figure. Such shapes possess three dimensions, i.e., length, breadth, and height (or depth). Most of the 3D shapes have faces that are 2D in nature, while some others have curved surfaces. The major 3D figures are cone, cylinder, cuboid, cube, and sphere.
There are specific terms used for describing the key attributes of 2D and 3D shapes. While a few concepts are common to both kinds of figures, some parameters can be estimated exclusively for either 2D or 3D shapes.
- Perimeter:
Perimeter refers to the complete boundary length or outline of a given figure. It can be computed only for the plane figures. - Volume
Volume denotes the space occupied by a figure. The concept of volume applies to 3D shapes only. - Area
The area is calculated both for plane and solid figures. In the case of 2D shapes, it signifies the amount of surface taken up by the given closed region. On the other hand, the 3D shapes entail the idea of total surface area and curved surface areas explained as follows:Curved Surface Area (CSA):
CSA is measured for figures like the sphere and cone that have curved surfaces. It includes the area of the curved part, excluding the base or top.Lateral Surface Area (LSA):
LSA indicates the total area of all sides or lateral surfaces that surround the figure.Total Surface Area (TSA):
TSA is the sum of lateral and curved surface areas of a 3D shape.
The crucial specifications of 2D shapes are their perimeters and areas. The given table illustrates the relevant formulas for popular 2D figures:
| Figure | Perimeter | Area | Dimensions |
| Square | 4a | a2 | ‘a’ is the length of each side. |
| Rectangle | 2(b + l) | bl | ‘b’ & ‘l’ are the breadth and length of the rectangle. |
| Triangle | a + b + c | √[s(s−a)(s−b)(s−c)] | ‘a’, ‘b’, & ‘c’ refer to three sides of a triangle, and ‘s’ is its semi-perimeter. |
| Circle | 2πr | πr2 | ‘r’ is the radius of a circle. |
| Parallelogram | 2(l + b) | hb | ‘l’ & ‘b’ are the length of sides, and ‘h’ is the height of the parallelogram. |
Square, rectangle, and parallelogram are basically quadrilateral types of polygons that differ from each other based on the angles and relationship between different sides. However, all these three quadrilaterals have two pairs of parallel sides. This is not the case with the trapezium.
A trapezium contains only one pair of parallel sides while the other two are non-parallel. The area of a trapezium is determined as ½ h(a + c), where ‘a’ and ‘c’ denote the length of parallel sides and ‘h’ is the height of the trapezium.
Apart from the simple 2D shapes mentioned above, certain complex questions may be asked regarding the area of a polygon and semi-circle, etc.
3D figures are specified using area and volume. Listed below are the important formulas for common 3D shapes:
| Figure | Total Surface Area | Volume | Dimensions |
| Cube | 6a2 | a3 | ‘a’ refers to the length of each side. |
| Cuboid | 2(lb + hl + bh) | lbh | ‘l’, ‘b’, and ‘h’ are the length, width, and height, respectively. |
| Cylinder | 2πr2+ 2πrh | πr2h | ‘r’ is the radius, and ‘h’ is the height of the cylinder. |
| Sphere | 4πr2 | (4/3)πr3 | ‘r’ is the radius of the sphere. |
| Cone | πr(r + l) | (1/3)πr2h | ‘r’, ‘h’, and ‘l’ refer to the radius, height, and side length of the cone. |
Questions related to the volume of a cube, cuboid and cylinder frequently feature in the examinations due to the higher significance and applications of these figures in daily life.
The concepts and formulas of mensuration can be effectively used to measure different parameters related to 2D and 3D figures. While 2D shapes help understand the fundamentals of the topic, the 3D structures form a further extension of the same. Using the above-mentioned formulas, you can easily solve the simple as well as mixed-type questions of the topic.
1. What are the applications of mensuration?
The formulas of mensuration find extensive use in construction, agriculture, design, and similar other industries.
2. What are the units for area and volume?
Since the area is written in square units, its SI unit is square metres (m2). On the other hand, you can carry out the measurement of volume using a basic unit of litre (ℓ) or SI units of cubic meter (m3).
3. What is the difference between volume and capacity?
Volume is the total space occupied by a solid object, whereas capacity is the quantity of any liquid/gas/solid that an object can hold. Capacity is mostly measured in litres.
4. What is the curved surface area of a sphere?
The curved surface area of a sphere is the same as its total surface area, i.e., 4πr2, where ‘r’ is the radius of the sphere.
5. What is the formula for the lateral surface area of a cube?
LSA of a cube = 4a2, where ‘a’ is the length of each side.
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