Mensuration – Definition, Introduction, Formulas, Solved Problems

In Maths Mensuration is nothing but a measurement of 2-D and 3-D Geometrical Figures. Mensuration is the study of the measurement of shapes and figures. We can measure the area, perimeter, and volume of geometrical shapes such as Cube, Cylinder, Cone, Cuboid, Sphere, and so on.

Keep reading this page to learn deeply about the mensuration. We can solve the problems easily, if and only we know the formulas of the particular shape or figure. This article helps to learn the mensuration formulae with examples. Learn the difference between the 2-D and 3-D shapes from here. Understand the concept of Mensuration by using various formulas.

Definition of Mensuration

Mensuration is the theory of measurement. It is the branch of mathematics that is used for the measurement of various figures like the cube, cuboid, square, rectangle, cylinder, etc. We can measure the 2 Dimensional and 3 Dimensional figures in the form of Area, Perimeter, Surface Area, Volume, etc.

What is a 2-D Shape?

The shape or figure with two dimensions like length and width is known as the 2-D shape. An example of a 2-D figure is a Square, Rectangle, Triangle, Parallelogram, Trapezium, Rhombus, etc. We can measure the 2-D shapes in the form of Area (A) and Perimeter (P).

What is 3-D Shape?

The shape with more or than two dimensions such as length, width, and height then it is known as 3-D figures. Examples of 3-Dimensional figures are Cube, Cuboid, Sphere, Cylinder, Cone, etc. The 3D figure is determined in the form of Total Surface Area (TSA), Lateral Surface Area (LSA), Curved Surface Area (CSA), and Volume (V).

Introduction to Mensuration

The important terminologies that are used in mensuration are Area, Perimeter, Volume, TSA, CSA, LSA.

  • Area: The Area is an extent of two-dimensional figures that measure the space occupied by the closed figure. The units for Area is square units. The abbreviation for Area is A.
  • Perimeter: The perimeter is used to measure the boundary of the closed planar figure. The units for Perimeter is cm or m. The abbreviation for Perimeter is P.
  • Total Surface Area: The total surface area is the combination or sum of both lateral surface area and curved surface area. The units for the total surface area is square cm or m. The abbreviation for the total surface area is TSA.
  • Lateral Surface Area: It is the measure of all sides of the object excluding top and base. The units for the lateral surface area is square cm or m. The abbreviation for the lateral surface area is LSA.
  • Curved Surface Area: The area of a curved surface is called a Curved Surface Area. The units of the curved surface area are square cm or m. The abbreviation for the curved surface area is CSA.
  • Volume: Volume is the measure of the three dimensional closed surfaces. The units for volume is cubic cm or m. The abbreviation for Volume is V.

Mensuration Formulas for 2-D Figures

Check out the formulas of 2-dimensional figures from here. By using these mensuration formulae students can easily solve the problems of 2D figures.

1. Rectangle:

  • Area = length × width
  • Perimeter = 2(l + w)

2. Square:

  • Area = side × side
  • Perimeter = 4 × side

3. Circle:

  • Area = Πr²
  • Circumference = 2Πr
  • Diameter = 2r

4. Triangle:

  • Area = 1/2 × base × height
  • Perimeter = a + b + c

5. Isosceles Triangle:

  • Area = 1/2 × base × height
  • Perimeter = 2 × (a + b)

6. Scalene Triangle:

  • Area = 1/2 × base × height
  • Perimeter = a + b + c

7. Right Angled Triangle:

  • Area = 1/2 × base × height
  • Perimeter = b + h + hypotenuse
  • Hypotenuse c = a²+b²

8. Parallelogram:

  • Area = a × b
  • Perimeter = 2(l + b)

9. Rhombus:

  • Area = 1/2 × d1 × d2
  • Perimeter = 4 × side

10. Trapezium:

  • Area = 1/2 × h(a + b)
  • Perimeter = a + b + c + d

11. Equilateral Triangle:

  • Area = √3/4 × a²
  • Perimeter = 3a

Mensuration Formulas of 3D Figures

The list of the mensuration formulae for 3-dimensional shapes is given below. Learn the relationship between the various parameters from here.

1. Cube:

  • Lateral Surface Area = 4a²
  • Total Surface Area = 6a²
  • Volume = a³

2. Cuboid:

  • Lateral Surface Area = 2h(l + b)
  • Total Surface Area = 2(lb + bh + lh)
  • Volume = length × breadth × height

3. Cylinder:

  • Lateral Surface Area = 2Πrh
  • Total Surface Area = 2Πrh + 2Πr²
  • Volume = Πr²h

4. Cone:

  • Lateral Surface Area = Πrl
  • Total Surface Area = Πr(r + l)
  • Volume = 1/3 Πr²h

5. Sphere:

  • Lateral Surface Area = 4Πr²
  • Total Surface Area = 4Πr²
  • Volume = (4/3)Πr³

6. Hemisphere:

  • Lateral Surface Area = 2Πr²
  • Total Surface Area = 3Πr²
  • Volume = (2/3)Πr³

Solved Problems on Mensuration

Here are some questions that help you to understand the concept of Mensuration. Use the Mensuration formulas to solve the problems.

1. Find the Length of the Rectangle whose Perimeter is 24 cm and Width is 3 cm?

Solution:

Given that,
Perimeter = 24 cm
Width = 3 cm
Perimeter of the rectangle = 2(l + w)
24 cm = 2(l + 3 cm)
2l + 6 = 24
2l + 6 = 24
2l = 24 – 6 = 18
2l = 18
l = 9 cm
Thus length of the rectangle = 9 cm

2. Calculate the volume of the Cuboid whole base area is 60 cm² and height is 5 cm.

Solution:

Given,
Base area = 60 cm²
Height = 5 cm
Volume of the Cuboid = base area × height
V = 60 cm² × 5 cm
V = 300 cm³
Thus the volume of the cuboid is 300 cm³.

3. Find the area of the Cube whose side is 10 centimeters.

Solution:

Given, side = 10 cm
Lateral Surface Area = 4a²
LSA = 4 × 10 × 10 = 400 cm²
Total Surface Area = 6a²
= 6 × 10 × 10 = 600 cm²
Volume of the cube = a³
V = 10 × 10 × 10 = 1000 cm³
Therefore the volume of the cube is 1000 cubic centimeters.

4. What is the lateral surface area of the sphere if the radius is 5 cm.

Solution:

Given,
The radius of the sphere = 5 cm
The formula for LSA of sphere = 4Πr²
Π = 3.14 or 22/7
LSA = 4 × 3.14 × 5 cm × 5 cm
LSA = 314 sq. cm
Thus the lateral surface area of the sphere is 314 sq. cm

5. What is the area of the parallelogram if the base is 15 cm and height is 10 cm.

Solution:

Given, Base = 15 cm
Height = 10 cm
We know that,
Area of parallelogram = bh
A = 15 cm × 10 cm
A = 150 sq. cm
Therefore the area of the parallelogram is 150 sq. cm.

FAQs on Mensuration

1. What is the use of Mensuration?

Mensuration is used to find the length, area, perimeter, and volume of the geometric figures.

2. What is the difference between 2D and 3D figures?

In 2D we can measure the area and perimeter. In 3D we can measure LSA, TSA, and Volume.

3. What is Mensuration in Math?

Mensuration is the branch of mathematics that studies the theory of measurement of 2D and 3D geometric figures or shapes.