Mensuration :The branch of mathematics that deals with measuring is known as Mensuration. It was first used for land surveys and other civic works in Egypt. The father of mensuration is Archimedes. Mensuration is a discipline of mathematics that studies the measurements of various geometrical forms and their areas, perimeters, and volumes, among other things. We’ll go through mensuration in-depth in this post, complete with formulas and examples. Continue reading to learn more.

Introduction to Mensuration: Mensuration Shapes

The ancient Egyptians first used the methods of mathematics for surveying and levelling the land. These methods of mathematics used here are the mensuration formulas. Understanding the mensuration meaning is crucial for every class 8 student as it is an important concept in NCERT Class 8 Maths book.

Measurements provide the height, width, depth, perimeter, area, and volume of a single object or group of objects. Mensuration outlines various geometrical shapes, such as two-dimensional shapes, their properties, and the formulas used.

Mensuration shapes apply to two-dimensional and three-dimensional shapes. They are

Mensuration shapes one-dimensional Figures:

1. Line
2. Line segment
3. Ray

Mensuration shapes two-dimensional figures:

1. Square
2. Rectangle
3. Triangle
4. Parallelogram
5. Trapezium
6. Circle

Mensuration shapes three-dimensional figures:

1. Cube
2. Cuboid
3. Cylinder
4. Cone
5. Sphere
6. Hemi-sphere

Mensuration shapes One-Dimensional (1D) Figures

The figures or shapes with only one measurement are known as one-dimensional figures (1D figures). 

A breadthless length is called the line. The line segment is the part of the line connected between two points with a fixed length. A ray is the part of the line starting from one point and extending infinitely in one direction.

Mensuration shapes Two-Dimensional (2D) Figures

The geometrical figures or shapes with only two measurements are called two-dimensional figures and are termed 2D figures.  

The 2D figures have only two dimensions, mainly length and breadth. The two-dimensional figure doesn’t have height or depth. We can measure the perimeter and area of the two-dimensional figures. The perimeter of two-dimensional figures is measured in \({\rm{m}},\,{\rm{cm}},\,{\rm{mm}},\,{\rm{km}},\) inches and feet. And, the area of the two-dimensional figures are measured in \({{\rm{m}}^{\rm{2}}}{\rm{,\;c}}{{\rm{m}}^{\rm{2}}}{\rm{,\;k}}{{\rm{m}}^{\rm{2}}}\) square feet.  

The various two-dimensional figures used in general are

1. Square
2. Rectangle
3. Parallelogram
4. Trapezium
5. Circle
6. Triangles

Mensuration shapes Three-Dimensional (3D) Figures

The geometrical figures or shapes with three measurements are three-dimensional, termed 3D figures.

The 3D figures or solids have three measurements: length, breadth, height, or depth. We can see more solids such as books, pencils, cylinders, balls, etc. are examples of three-dimensional figures in our daily lives. We can measure the surface areas (curved or lateral surface area and total surface area) and volume.

The surface area is generally measured in square units like \({{\rm{m}}^{\rm{2}}}{\rm{,\;c}}{{\rm{m}}^{\rm{2}}}{\rm{,\;k}}{{\rm{m}}^{\rm{2}}}\) square hectares, square feet. The volume of the three-dimensional figures is measured in cubic units like \({\rm{c}}{{\rm{m}}^{\rm{3}}}{\rm{,}}{{\rm{m}}^{\rm{3}}}{\rm{,\;k}}{{\rm{m}}^{\rm{3}}}{\rm{,}}\) cubic hectares etc.

The general three-dimensional figures used are

1. Cube
2. Cuboid
3. Cylinder
4. Sphere
5. Hemi-sphere
6. Cone

Learn All Important Mensuration Formulas

Important Terminologies Used in Mensuration

The various terms associated with mensuration are discussed below:

1. Perimeter:
The measure of the continuous length of the boundary of the geometrical figure is called the perimeter of the figure. It is measured in units of \({\rm{m,cm,}}\) inches etc.

2. Area:
The amount of space or region covered by the geometrical figure is called the area of the figure. It is measured in square units like \({{\rm{m}}^{\rm{2}}}{\rm{,\;c}}{{\rm{m}}^{\rm{2}}}{\rm{,\;k}}{{\rm{m}}^{\rm{2}}}\) etc.

3. Lateral surface area:
The area or region covered by the laterals (line segments) of the geometrical figure is called the lateral surface area, and it is represented as LSA.

4. Curved surface area:
The area or region covered by the curved faces of the geometrical figure is called the curved surface area, and it is represented as CSA.

5. Total Surface area:
The area or region covered by the entire figure is called the total surface area, and it is represented as TSA.

6. Volume:
The space inside the three-dimensional figure is called the volume of the figure, and it is measured in cubic units like \({\rm{c}}{{\rm{m}}^{\rm{3}}}{\rm{,}}{{\rm{m}}^{\rm{3}}}{\rm{,\;k}}{{\rm{m}}^{\rm{3}}}{\rm{,}}\) cubic hectares etc.

Mensuration Formulas for 2D Figures

The formulas are used to measure the two-dimensional objects mathematically.

Square

A quadrilateral with all four sides and angles are equal is called square.

Perimeter \( = 4 \times {\rm{side}} = 4\,a\)

Area \({\rm{ = side \times side = }}{a^2}\)

Rectangle

Consider a rectangle with length \((l)\) and breadth \((b)\):

Perimeter \({\rm{ = 2(length + breadth) = 2(l + b)}}\)

Area \({\rm{ = length \times breadth = l \times b}}\)

Parallelogram

Consider the parallelogram \(ABCD,\) with base \((b)\) and height \((h)\):

Perimeter \( = AB + BC + CD + AD\)

Area \({\rm{ = base \times height = b \times h}}\)

Rhombus

Consider the rhombus with the length of the diagonals are \({d_1}\) and \({d_2}\): 

Perimeter \(=\) sum of all four sides.

Area \( = \frac{1}{2} \times {d_1} \times {d_2}\)

Trapezium

Consider the trapezium with the length of parallel sides are \(a\) and \(c,\) and distance between the parallel sides is \(h\):

Perimeter \( = a + b + c + d\)

Area \( = \frac{1}{2} \times (a + c) \times h\)

Scalene Triangle

Consider a scalene triangle with sides of

lengths \(a, b, c\) : Perimeter \(=a+b+c\)

Area \(=\sqrt{s(s-a)(s-b)(s-c)}\)

Here, \(s=\frac{a+b+c}{2}\)

Equilateral Triangle

Consider an equilateral triangle with side \(” a “\):

Perimeter \(=3a\)

Area \(=\frac{\sqrt{3}}{4} a^{2}\) 

Isosceles Triangle

Consider an isosceles triangle with an equal length of \(” a “\) and base length \(” b”\), height \(“h.”\)

Perimeter \(=2a+b\)

Area \(=\frac{1}{2} \times b \times h\)

Or area \(= \frac{1}{2} \times b \times \sqrt {\left( {{a^2} – \frac{{{b^2}}}{4}} \right)} \)

Right-Angled Triangle

Consider a right-angled triangle  of base \(“b”\) and height \(“h.”\)

Perimeter \( = b + h + {\rm{hypotenuse}}\)

Area \(= \frac{1}{2} \times b \times h\)

Circle

Consider a circle with radius \(“r”:\)

Circumference or perimeter \(=2 \pi r\)

Area \(=\pi r^{2}\)

Semi Circle

Consider a semi-circle with radius \(“r”:\)

Circumference or perimeter \( = \pi r + 2r\)

Area \( = \frac{\pi }{2}{r^2}\)

Mensuration Formulas for Three-Dimensional Figures

The measurements of common three-dimensional figures are provided here:

Cube

Consider a cube of side \(“a”\) units

1. Lateral surface area (LSA) \(=4 a^{2}\)
2. Total surface area (TSA) \(=6 a^{2}\)
3. Volume \(=a^{3}\)

Cuboid

Consider a cuboid of length \(“l”\), breadth \(“b”\) and height \(“h”\).

1. Lateral surface area (LSA) \( = 2h(l + b)\)
2. Total surface area (TSA) \( = 2(lb + bh + hl)\)
3. Volume \( = l \times b \times h\)

Right Circular Cylinder

Consider a right circular cylinder of radius \(“r”\) and height \(“h”\).

1. Curved surface area (CSA) \(=2 \pi \mathrm{rh}\)
2. Total surface area (TSA) \(=2 \pi r(r+h)\)
3. Volume \(=\pi r^{2} h\)

Hollow Cylinder

Consider a hollow cylinder of inner radius \(“r”\), outer radius \(“R”\) and height \(“h”\).

1. Curved surface area (C.S.A) \(=2 \pi(r+R) h\)
2. Total surface area (T.S.A) \(=2 \pi(r+R)(R+h-r)\)
3. Volume \(=\pi h\left(R^{2}-r^{2}\right)\)

Right Circular Cone

Consider a cone with radius \(“r”\), height \(“h”\) and slant height \(“l”\).

1. Slant height \((l)=\sqrt{r^{2}+h^{2}}\)
2. Curved surface area (CSA) \(=\pi r l\)
3. Total surface area (TSA) \(=\pi r(r+l)\)
4. Volume \(=\frac{1}{3} \times \pi r^{2} h\)

Frustum of Cone

Consider a frustum of a cone with base radius \(R\) and top radius \(r\) with height \(“h”\) and slant height \(“l”\).

1. Slant height \((l)=\sqrt{(R-r)^{2}+h^{2}}\)
2. Curved surface area (CSA) \(=\pi l(R+r)\)
3. Total surface area (TSA) \(=\pi l(r+R)+\pi r^{2}+\pi R^{2}\)
4. Volume \(=\frac{1}{3} \times \pi h\left(r^{2}+r R+R^{2}\right)\)

Sphere

Consider a sphere with radius \(“r”\)

1. Surface area \( = 4\pi {r^2}\)
2. Volume \( = \frac{4}{3}\pi {r^3}\)

Hemi Sphere

Consider a hemisphere of radius \(“r”\)

1. Curved surface area (CSA) \( = 2\pi {r^2}\)
2. Total surface area (TSA) \( = 3\pi {r^2}\)
3. Volume \( = \frac{2}{3}\pi {r^3}\)

Solved Examples – Mensuration

Q.1. The cylindrical milk bottle is of radius \({\rm{2\,cm}}\), and height is \({\rm{5\,cm}}\). Find the capacity of milk in the given bottle? (Use: \(\pi = \frac{{22}}{7},\))?

Ans: Given the radius \((r) = 2\;{\rm{cm}}\) and height \((h) = 5\;{\rm{cm}}\)
We know that the capacity of the cylindrical bottle is known as the volume of the cylinder.
The volume of the cylinder with radius \(‘r’\) and height \(‘h’\) is \(\pi {r^2}h\).
\( = \frac{{22}}{7} \times {2^2} \times 5\)
\( = \frac{{22}}{7} \times 20\)
\( = \frac{{440}}{7}\)
\( = 62.86\;{\rm{c}}{{\rm{m}}^3}\)
Hence, the capacity of the given milk bottle is \(62.86\;{\rm{c}}{{\rm{m}}^3}\) (Approx).

Q.2. Find the triangle area with sides is \({\rm{3\,cm,3\,cm}}\) and \({\rm{4\,cm}}\)?
Ans: Given, lengths of the sides of a triangle are \(3\;{\rm{cm}},3\;{\rm{cm}},4\;{\rm{cm}}{\rm{.}}\)

So, \(a{\rm{ = 3\,cm}}\) and \(b{\rm{ = 4\,cm}}\)
The area of the isosceles triangle is given by \(\frac{1}{2} \times b \times \sqrt {\left( {{a^2} – \frac{{{b^2}}}{4}} \right)} \)
\( = \frac{1}{2} \times 4 \times \sqrt {{3^2} – \frac{{{4^2}}}{4}} \)
\( = 2 \times \sqrt {9 – 4} \)
\( = 2\sqrt 5 \;{\rm{c}}{{\rm{m}}^2}\)
Hence, the area of the given isosceles triangle is \(2\sqrt 5 \;{\rm{c}}{{\rm{m}}^2}\)

Q.3. Find the area and perimeter of the square whose sides measure \({\rm{6\,cm}}\)?
Ans: Given the square of the side is \({\rm{6\,cm}}\).

We know that the area of the square is given by [{\rm{side}} \times {\rm{side}}]
Area \(=6 \times 6=6^{2}=36 \mathrm{~cm}^{2}\)
Perimeter \(=4 \times \rm{side}} =4 \times 6=24 \mathrm{~cm}\)
Hence, the area and perimeter of the given square is \(36 \mathrm{~cm}^{2}\) and \(24 \mathrm{~cm}\).

Q.4. Find the volume of the tent, which is in the form of a right circular cone, with radius and height are \({\rm{24}}\,{\rm{m}}\) and \({\rm{10}}\,{\rm{m}}\) respectively?
Ans: Given that, a tent is in the form of a right circular cone.
And, the radius  \((r)={\rm{24}}\,{\rm{m}}\), the height \((h)={\rm{10}}\,{\rm{m}}\)

We know that volume of the right circular cone is \(\frac{1}{3} \pi r^{2} h\)
\(V = \frac{1}{3} \times \frac{{22}}{7} \times {24^2} \times 10\)
\(\Rightarrow V=\frac{42,240}{7} {{\rm{m}}^3}\)
Hence, the volume of the tent is \(\frac{42,240}{7} {{\rm{m}}^3}\)

Q.5. A \(15\) inches (diameter) pizza is served to customers. Calculate its circumference?

Ans: Diameter of pizza \((d)=15\) inches
The circumference of the circle in terms of diameter is \(C=\pi d\)
\(C=\pi \times 15=15 \pi\) inches 
Hence, the circumference of a pizza is \(15\pi \) inches.

Summary

In this article, we have studied the definition of mensuration, which deals with the study of measurements of \(2D\) and \(3D\) figures. We have also studied the meaning of one-dimensional, two-dimensional, and three-dimensional figures meanings.

This article also gives the various formulas used in two-dimensional figures such as square, rectangle, triangle, parallelogram, trapezium, etc. And three-dimensional figures such as cubes, cuboids, cylinders, cones, spheres, etc, with the help of solved examples, to understand the concepts easily.

NCERT Solutions for Mensuration Chapter

Frequently Asked Questions (FAQs) – Mensuration

The answers to the most commonly asked questions about Mensuration are provided here:

Q.1. What is mensuration in math?
Ans: Mensuration is the special branch of mathematics, that deals with the study of measurements of various geometrical figures and their areas, perimeters, volumes, etc.

Q.2. What are the types of mensuration?
Ans: Two types of mensuration are 2D and 3D mensuration. Mensuration applies to two-dimensional (2D) figures like squares, rectangles, triangles, parallelograms, trapezium, etc. It applies to three-dimensional (3D) figures like a cube, cuboid, cylinder, cone, sphere, etc.

Q.3. Who invented mensuration?
Ans: Mensuration was first introduced in Egypt for making land surveying and other civil works. Archimedes is the father of mensuration. 

Q.4. What is the use of mensuration?
Ans: The mensuration is used to calculate the perimeter, area, and volume of two-dimensional and three-dimensional figures.

Q.5. What is the volume in mensuration?
Ans: The space inside the three-dimensional figure is called the volume of the figure, and it is measured in cubic units like \({\rm{c}}{{\rm{m}}^{\rm{3}}}{\rm{,}}{{\rm{m}}^{\rm{3}}}{\rm{,\;k}}{{\rm{m}}^{\rm{3}}}{\rm{,}}\) cubic hectares etc.

We hope this detailed article on mensuration helped you in your studies. If you have any doubts, queries, or suggestions regarding this article, feel to ask us in the comment section and we will be more than happy to assist you. Happy learning!