Mensuration is a branch of mathematics concerned with the calculation of geometric figures and their parameters such as weight, volume, form, surface area, lateral surface area, and so on.
This article will help you to learn about mensuration in a very easy manner. Principles of mensuration along with all the essential formulas of mensuration are discussed here.
For a better understanding of these ideas, the properties of various geometric forms and the accompanying figures are also shown. Mensuration is a branch of mathematics that deals with the scale, volume, or area of 2D or 3D shapes.
Mensuration Definition Mensuration in simple terms means measurement of different geometrical shapes.
These shapes can exist in 2 ways:
Two-Dimensional Shapes – circle, triangle, square, etc. Three-Dimensional Shapes – cube, cuboid, cone, etc.
Difference Between 2D and 3D shapes
Any shape is 2D if it is bound by three or more straight lines in a plane. A shape is a three-dimensional shape if there are several surfaces or planes around it.
There is no height or depth in these shapes. In contrast to 2D forms, these are sometimes known as solid shapes and have height or depth.
These shapes just have length and width as their dimensions. Since they have depth (or height), breadth, and length, they are referred to as three-dimensional objects.
We can calculate their perimeter and area. Their volume, curved surface area, lateral surface area, or total surface area can all be calculated.
Mensuration Terminologies Here is the list of terms you will come across in mensuration class. We have provided the term, it’s abbreviation, unit and definition for easy understanding.
Area A m 2 or cm 2 The surface that the closed form covers is known as the area.
Perimeter P cm or m A perimeter is the length of the continuous line that encircles the specified figure.
Volume V cm 3 or m 3 A 3D shape’s space is referred to as its volume.
Curved Surface Area CSA m 2 or cm 2 The overall area is known as a Curved surface area if there is a curved surface. Example: Sphere
Lateral Surface area LSA m 2 or cm 2 The term “Lateral Surface area” refers to the combined area of all lateral surfaces that encircle the provided figure.
Total Surface Area TSA m 2 or cm 2 The total surface area is the total of all the curved and lateral surface areas.
Square Unit – m 2 or cm 2 A square unit is the area that a square of side one unit covers.
Cube Unit – m 3 or cm 3 The space taken up by a cube with a single side.
Mensuration of 2D shapes Mensuration of two-dimensional figures measure things such as area, perimeter, and so on.
The form or figure having two dimensions, such as length and breadth, is referred to as a 2-D form.
For example , square, rectangle, triangle, parallelogram, trapezium, rhombus, and so on. We can measure 2-D forms using Area (A) and Perimeter (P) as discussed below:
Area of Trapezium Area of Polygons Area of General Quadrilateral Heron’s Formula Applications of Heron’s Formula Area of 2D Shapes Perimeter of circular figures Areas of sector and segment of a circle Areas of combination of plane figures
Mensuration of 3D shapes Mensuration is also concerned with the measuring of three-dimensional solids in terms of total surface area, lateral/curved surface area, and volume.
3D figures are those that have more than two dimensions, such as length, breadth, and height.
Cube, Cuboid, Sphere, Cylinder, Cone, and other three-dimensional forms are examples.
Total Surface Area, Lateral Surface Area, Curved Surface Area, and Volume are used to calculate the 3D figure are discussed in the articles below:
Surface Area of Cube, Cuboid, and Cylinder Volume of Cube, Cuboid, and Cylinder Volume and Capacity Surface Area of 3D Shapes Volumes of Cubes and Cuboids Surface Areas and Volumes Volumes of a combination of solids Conversion of solids Frustum of a Cone Section of a Cone Conic Section Parabola Ellipse Hyperbola
Shape Area (Square units) Perimeter (units) Figure
Square a 2 4a Square dimensions
Rectangle l × b 2 (l + b) rectangle dimensions
Circle πr 2 2 π r circle with radius
Scalene Triangle √[s(s−a)(s−b)(s−c)], Where, s = (a+b+c)/2 a+b+c Scalene triangle dimensions
Isosceles Triangle ½ × b × h 2a + b isosceles triangle dimensions
Equilateral triangle (√3/4) × a 2 3a Equilateral triangle dimensions
Right Angle Triangle ½ × b × h
Right Angle Triangle dimensions
Rhombus ½ × d1 × d2 4 × side Â
Parallelogram b × h 2(l+b) Parallelogram dimensions
Trapezium ½ h(a+c) a+b+c+d Trapezium dimensions
Shape Volume (Cubic units) Curved Surface Area (CSA) or Lateral Surface Area (LSA) (Square units) Total Surface Area (TSA) (Square units) Figure
Cube a 3 LSA = 4 a 2 6 a 2 cube dimensions
Cuboid l × b × h LSA = 2h(l + b) 2 (lb +bh +hl) cuboid dimensions
Sphere (4/3) π r 3 4 π r 2 4 π r 2 sphere dimensions
Hemisphere (⅔) π r 3 2 π r 2 3 π r 2 hemisphere dimensions
Cylinder π r 2 h 2π r h 2πrh + 2πr 2 cylinder dimensions
Cone (⅓) π r 2 h π r l πr (r + l)
Solved Problems on Mensuration Problem 1: Find the volume of a cone if the radius of its base is 1.5 cm and its perpendicular height is 5 cm.
Solution:
Radius of the cone, r = 1.5 cm
Height of the cone, h = 5 cm
∴ Volume of the cone, V = 13πr 2 h=13×227×(1.5) 2 ×5= 11.79 cm 3
Thus, the volume of the cone is 11.79 cm 3 .Â
Problem 2: The dimensions of a cuboid are 44 cm, 21 cm, 12 cm. It is melted and a cone of height 24 cm is made. Find the radius of its base.
Solution:
The dimensions of the cuboid are 44 cm, 21 cm and 12 cm.
Let the radius of the cone be r cm.
Height of the cone, h = 24 cm
It is given that cuboid is melted to form a cone.
∴ Volume of metal in cone = Volume of metal in cuboid
⇒(1/3)Ï€r 2 h=44×21×12       Â
 (Volume of cuboid=Length×Breadth×Height)
⇒(1/3)×(22/7)×r 2 ×24=44×21×12
⇒r= √(44×21×12×21) / (22×24)
=21 cm
Thus, the radius of the base of cone is 21 cm. Â
Problem 3: The radii of two circular ends of frustum shape bucket are 14 cm and 7 cm. The height of the bucket is 30 cm. How many liters of water can it hold? (1 litre = 1000 cm 3 )
Solution:
Radius of one circular end, r1 = 14 cm
Radius of other circular end, r2 = 7 cm
Height of the bucket, h = 30 cm
∴ Volume of water in the bucket = Volume of frustum of cone =(1/3)πh(r 1 2 +r 1 r 2 +r 2 2 )
=13×22/7×30×(142+14×7+72)
=13×22/7×30×343=10780 cm 3
=107801000=10.780 L
Thus, the bucket can hold 10.780 litres of water.
Problem 4: How many bricks each measuring 64 cm x 11.25 cm x 6 cm. will be needed to build a wall measuring 8m x 3m x 22.5m?
Solution:
Volume of wall = V1 Volume of Brick= V2 Number of Bricks =V1/V2 =(8m*3m*22.5)/(64cm*11.25cm*6cm) =(800*300*22.5)/(64*11.25*6) = 125000
We have covered all the concept of mensuration in maths. We have also discussed the mensuration terminologies, 2D and 3D shapes with their mensuration formulas. You can try solving the samples questions to try your abilities.
Mensuration is very important concept of Geometry, and also has many real life applications like in construction, engineering, land survey etc.
FAQs on Mensuration Q1: What is Mensuration?
Mensuration is a branch of mathematics concerned with calculation of geometric figures and their parameters such as weight, volume, form, surface area, lateral surface area, and so on.
Q2: What are 2D and 3D shapes?
Any shape is considered to be 2D if it is bound by three or more straight lines in a plane whereas a shape is a three-dimensional shape if there are several surfaces or planes around it.
Q3: What is the formula for the area of a cylinder?
Lateral or Curved Surface area of a cylinder = 2Ï€ r h
Total Surface Area of a cylinder = 2Ï€rh + 2Ï€r 2
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Last Updated :
18 Dec, 2023
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