Mensuration in Maths | Formulas for 2D and 3D Shapes, Examples

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.

Let’s learn about all the mensuration formulas in maths.

Mensuration Meaning

Mensuration is the branch of mathematics that deals with the measurement of various geometric figures and shapes. This includes calculating areas, volumes, and perimeters of two-dimensional shapes like squares, rectangles, circles, and triangles, as well as three-dimensional figures like cubes, cylinders, spheres, and cones.

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

2-Dimensional vs 3-Dimensional Shapes
2D Shape	3D Shape
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.

Terms	Abbreviation	Unit	Definition
Area	A	m2 or cm2	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	cm3 or m3	A 3D shape’s space is referred to as its volume.
Curved Surface Area	CSA	m2 or cm2	The overall area is known as a Curved surface area if there is a curved surface. Example: Sphere
Lateral Surface area	LSA	m2 or cm2	The term “Lateral Surface area” refers to the combined area of all lateral surfaces that encircle the provided figure.
Total Surface Area	TSA	m2 or cm2	The total surface area is the total of all the curved and lateral surface areas.
Square Unit	–	m2 or cm2	A square unit is the area that a square of side one unit covers.
Cube Unit	–	m3 or cm3	The space taken up by a cube with a single side.

Mensuration Formula For 2D Shapes

The following table provides a list of all mensuration formulas for 2D shapes:

Shape	Area (Square units)	Perimeter (units)	Figure
Square	a2	4a	Square dimensions
Rectangle	l × b	2 (l + b)	rectangle dimensions
Circle	πr2	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) × a2	3a	Equilateral triangle dimensions
Right Angle Triangle	½ × b × h	b + hypotenuse + h	Right Angle Triangle dimensions
Rhombus	½ × d1 × d2	4 × side
Parallelograms	b × h	2(l+b)	Parallelogram dimensions
Trapezium	½ h(a+c)	a+b+c+d	Trapezium dimensions

Learn More:

• 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 Formula for 3D Shapes

The following table provides a list of all mensuration formulas for 3D shapes:

Shape	Volume (Cubic units)	Curved Surface Area (CSA) or Lateral Surface Area (LSA) (Square units)	Total Surface Area (TSA) (Square units)	Figure
Cube	a3	LSA = 4 a2	6 a2	cube dimensions
Cuboids	l × b × h	LSA = 2h(l + b)	2 (lb +bh +hl)	cuboid dimensions
Sphere	(4/3) π r3	4 π r2	4 π r2	sphere dimensions
Hemisphere	(⅔) π r3	2 π r2	3 π r2	hemisphere dimensions
Cylinder	π r 2 h	2π r h	2πrh + 2πr2	cylinder dimensions
Cone	(⅓) π r2 h	π r l	πr (r + l)

Learn More :

• 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

Solved Problems on Mensuration

Let’s solve some example 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 = 13πr²h=13×227×(1.5)²×5= 11.79 cm³

Thus, the volume of the cone is 11.79 cm³. 

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)πr²h=44×21×12             

 (Volume of cuboid=Length×Breadth×Height)

⇒(1/3)×(22/7)×r²×24=44×21×12

⇒r= √(44×21×12×21) / (22×24)

=21 cm

Thus, the radius of the base of cone is 21 cm.
 

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 cm³).

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(r₁²+r₁r₂+r₂²)

=13×22/7×30×(142+14×7+72)

=13×22/7×30×343=10780 cm³

=107801000=10.780 L

Thus, the bucket can hold 10.780 litres of water.

FAQs On Mensuration

What is Mensuration?

Mensuration deals with the calculation of geometric figures and their parameters such as weight, volume, form, surface area, lateral surface area, and so on.

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.

What is Area of Cylinder Formula?

Lateral or Curved Surface area of a cylinder = 2π r h

Total Surface Area of a cylinder = 2πrh + 2πr²

What is TSA (Total Surface Area) of Sphere Formula?

Area of Sphere is given by the following formula :

A= 4 π r²

What is Volume of Cone Formula?

Volume of Cone is given by the following formula :

V= (⅓) π r² h

What is Area of Triangle Formula?

Area of Triangle is given by the following formula :

A= 1/2 ×b ×h

What is Area of Circle Formula?

Area of Circle is given by the following formula :

A= π r²

What is Volume of Cylinder Formula?

Volume of Cylinder is given by the following formula :

V= π r ² h