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November 21, 201839 Insightful Publications
Mensuration Formulas: Mensuration is a branch of mathematics that deals with the area, perimeter, volume, and surface area of various geometrical shapes. It is one of the most important chapters covered in high school Mathematics. Mensuration has immense practical applications in our daytoday life. It is, for this reason, advanced concepts related to mensuration are covered in higher grades. Knowing the basics of ‘what is mensuration?’ is the key to understand it.
It is also an important and scoring topic for competitive exams, like the Olympiads and NTSE. Mensuration problems are asked in various government job exams as well, like SSC, Banking, Insurance, etc. So it becomes very important for everyone to understand and memorize various mensuration formulas for all 2D and 3D geometrical figures. Keep reading to know more about what is mensuration and mensuration all formula.
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Mensuration is the branch of mathematics in which we study the surface area, volume, perimeter, length, breadth, and height of geometric shapes. Shapes can be 2D or 3D in nature. Let’s understand what are 2dimensional and 3dimensional shapes and what are the differences?
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In geometry, a twodimensional shape or 2D shape is defined as a flat plane figure or a shape that has only two dimensions. These shapes can be represented in a plane with Xaxis and Yaxis. Some common examples of 2D shapes are circle, square, rectangle, parallelogram, and rhombus.
A threedimensional shape or 3D shape is defined as a solid figure or an object that has three dimensions – length, breadth, and height. Threedimensional shapes can’t be represented on a plane. We need spatial representation for 3D shapes because they have an extra dimension as thickness or depth.
Let’s see the major differences between a 2D and a 3D shape:
2D Shape  3D Shape 

A 2D shape is surrounded by 3 or more straight lines that can be represented on a plane surface.  A 3D shape is surrounded by multiple surfaces or planes. They are represented spatially. 
2D shapes have only length and breadth, and no height.  3D shapes are solid figures and they have an extra dimension as depth or height. 
For 2D shapes, we measure the area and perimeter of figures.  For 3D shapes, we measure their volume, curved surface area, and total surface area. 
The table below shows the area and perimeter formulas of common 2D geometrical figures:
Mensuration Formulas for Different 2D Geometric Shapes  

Shape  Area Formula  Perimeter Formula  Figure 
Square 
\({{a}^{2}}\) 
\(4a\) 

Rectangle 
\(lw\) 
\(2(l+w)\) 

Rightangled Triangle 
\(\frac{1}{2}bh\) 
\(b+h+H\) 

Scalene Triangle 
\(\sqrt{s(sa)(sb)(sc)}\), 
\(a+b+c\) 

Isosceles Triangle 
\(\frac{1}{2}bh\) 
\(2a+b\) 

Equilateral Triangle 
\(\frac{\sqrt{3}}{4}{{a}^{2}}\) 
\(3a\) 

Parallelogram 
\(bh\) 
\(2(a+b)\) 

Trapezium 
\(\frac{1}{2}h(a+c)\) 
\(a+b+c+d\) 

Rhombus 
\(\frac{1}{2}{{d}_{1}}{{d}_{2}}\) 
\(4a\) 

Circle 
\(\pi{{r}^{2}}\) 
\(C=2\pi r\), 
Check the properties of different geometric shapes:
The table below shows the formulas for volume, curved surface area, and total surface area of common 3D geometrical figures:
Mensuration Formulas for Different 3D Geometric Shapes  

Shape  Volume Formula  Curved Surface Area Formula  Total Surface Area  Figure 
Cube 
\({{a}^{3}}\) 
\(4{{a}^{2}}\) 
\(6{{a}^{2}}\) 

Cuboid 
\(lbh\) 
\(2(l+b)h\) 
\(2(lb+bh+hl)\) 

Sphere 
\(\frac{4}{3}\pi{{r}^{3}}\) 
\(4\pi{{r}^{2}}\) 
\(4\pi{{r}^{2}}\) 

Hemisphere 
\(\frac{2}{3}\pi{{r}^{3}}\) 
\(2\pi{{r}^{2}}\) 
\(3\pi{{r}^{2}}\) 

Cylinder 
\(\pi{{r}^{2}}h\) 
\(2\pi rh\) 
\(2\pi rh+2\pi{{r}^{2}}\) 

Cone 
\(\frac{1}{3}\pi{{r}^{2}}h\) 
\(\pi rl\) 
\(\pi r(r+l)\) 
In mensuration, we come across different terminologies such as area, perimeter, surface area, volume, etc. We have provided definitions for all these terms so that you can be confident about all the mensuration concepts.
Here we have provided some mensuration problems with solutions.
Question 1: PQRS is a rectangle. The ratio of the sides PQ and QR is 3:1. If the length of the diagonal PR is 10 cm, then what is the area (in cm²) of the rectangle?
Solution: PQRS is a rectangle
PR = 10 given
PQ : QR = 3 : 1
In ∆PQR
9x² + x² = 100
10x² = 100
x = √10
Area of rectangle = 3x × 1x
= 3x²
= 3 × 10
= 30
Question 2: The height of a cone is 24 cm and the area of the base is 154 cm². What is the curved surface area (in cm²) of the cone?
Solution: Area of base = 154 cm²
πr² = 154
22/7×r^2
=154
r = 7
Height = 24
Radius = 7
Slant height (ℓ) = √(h²+r² )
ℓ =√(24²+7² )
ℓ=25
C.S.A. = πrℓ
= 22/7×7×25
C.S.A. ⇒ 550 cm²
Question 3: ABCD is a trapezium. Sides AB and CD are parallel to each other. AB = 6 cm, CD = 18 cm, BC = 8 cm and AD = 12 cm. A line parallel to AB divides the trapezium in two parts of equal perimeter. This line cuts BC at E and AD at F. If BE/EC = AF/FD, than what is the value of BE/EC?
Solution: Let BE = x then EC = 8 – x
BE/EC = AF/FD (Given)
Reverse the given condition & add 1 both side
EC/BE + 1 = FD/AF + 1
(EC+BE)/BE = (FD+AF)/AF
⇒ BC/BE = AD/AF … (i)
Put values in eq. (i)
→ 8/x = 12/AF
AF = 3x/2
Similarly, FD = 12–3x/2
Now perimeter FABE = FECD
FA + AB + BE + FE = EC + CD + DF + FE
3x/2 + 6 + x = 8 – x + 18 + (12–3x/2)
5x = 32
x = 32/5
=BE
Hence EC = 8 –32/5
= 8/5
∴ BE/EC = (32/5)/(8/5)
=4
Question 4: Find the area and perimeter of a square whose side is 10 cm.
Solution: Given: Side = a = 10 cm
Area of a square = a^{2 }square units
Substitute the value of “a” in the formula, we get
Area of a square = 10^{2}
A = 10 x 10 = 100
Therefore, the area of a square = 100 cm^{2}
The perimeter of a square = 4a units
P = 4 x 10 =40
Therefore, the perimeter of a square = 40 cm.
Question 5: Find out the height of a cylinder with a circular base of radius 70 cm and volume 154000 cubic cm.
Solution: Given, r= 70 cm
V= 154000 cm^{3}
Since formula is,
V = πr^{2}h
h = V/πr^{2}
= 154000/15400
=10
Hence, height of the cylinder is 10 cm.
Question 6: If the sides of a triangle are 26 cm, 24 cm, and 10 cm, what is its area?
Solution: The triangle with sides 26 cm, 24 cm and 10 cm is rightangled, where the hypotenuse is 26 cm.
Area of the triangle = 1/2 * 24 * 10 = 120 cm^{2}
Question 7: Find the area of a trapezium whose parallel sides are 20 cm and 18 cm long, and the distance between them is 15 cm.
Solution: Area of a trapezium = 1/2 (sum of parallel sides) * (perpendicular distance between them)
= 1/2 (20 + 18) * (15)
= 285 cm^{2}
Question 8: Find the area of a parallelogram with a base of 24 cm and a height of 16 cm.
Solution: Area of a parallelogram = base * height
= 24 * 16
= 384 cm2
At Embibe, you can solve mensuration practice questions for free:
Class 8 Mensuration Practice Questions 
Class 9 Mensuration Practice Questions 
Class 10 Mensuration Practice Questions 
Check other important Maths articles:
Students can find some general FAQs on the topic down below:
Q.1: What is the formula for mensuration?
Ans: Mensuration is commonly referred to as the study of geometry and the formulas that come under it involving the calculation of Area, Perimeter, Volume, and Surface Area of different types of 2D and 3D figures. For the full list of formulas, you can refer to this article.
Q.2: How can we remember mensuration formulas?
Ans: The best way to remember mensuration formulas would be by understanding area and perimeter concepts and then using the formula tables provided in this article. You can either take a printout of the page or bookmark it whenever you need it.
Q.3: Which is the easiest way of learning mensuration formulas?
Ans: The easiest way of learning mensuration formulas will be by taking the printout of the formulas provided in this article and sticking them near your study table so that you can revise them whenever you want or you can bookmark this page and visit for revision.
Q.4: Is there any difference between mensuration and geometry?
Ans: Mensuration deals with the calculation of perimeter, area, volume, and other parameters for 2D or 3D geometric shapes. Geometry is concerned with the properties and relations of points and lines of various shapes.
Q.5: What are 2D and 3D mensuration?
Ans: 2D mensuration deals with the area, perimeter, volume, and other parameters related to 2D geometric shapes such as square, rectangle, rhombus, circle, etc.
On the other hand, 3D mensuration is concerned with the calculation of volume, curved surface area, lateral surface area, and total surface area of 3D geometric shapes such as a sphere, cylinder, cone, etc.
Test Your Knowledge of Mensuration With Free Mock Test
Now you are provided with all the necessary information regarding different mensuration formulas. Practice more questions and master this concept. Students can make use of NCERT Solutions for Maths provided by Embibe for their exam preparation.
We hope this detailed list of mensuration formulas help you. If you have any queries, feel free to ask in the comment section below. We will get back to you at the earliest.