Mensuration

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. At GeeksforGeeks, you can 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 various geometric forms. These forms are available in two or three dimensions.

Table of Content

• Difference Between 2D and 3D Shapes
• Terminology
• Mensuration of 2D Shapes
• Mensuration of 3D Shapes
• Mensuration Formula for 2D Shapes
• Mensuration Formula for 3D Shapes
• Solved Problems
• FAQs

Difference Between 2D and 3D shapes

Mensuration in Maths-Terminology

Mensuration of 2D shapes

Mensuration is the measuring theory. It is the field of mathematics that is used to measure various figures such as cube, cuboid, square, rectangle, cylinder, and so on. Mensuration of two-dimensional figures 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:

1. Introduction to Mensuration
2. Area of Trapezium
3. Area of Polygons
4. Area of General Quadrilateral
5. Heron’s Formula
6. Applications of Heron’s Formula
7. Area of 2D Shapes
8. Perimeter of circular figures, 
9. Areas of sector and segment of a circle 
10. Areas of combination of plane figures
11. Circle

Mensuration of 3D shapes

Mensuration is 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:

1. Surface Area of Cube, Cuboid, and Cylinder
2. Volume of Cube, Cuboid, and Cylinder
3. Volume and Capacity
4. Surface Area of 3D Shapes
5. Volumes of Cubes and Cuboids
6. Surface Areas and Volumes
7. Volumes of a combination of solids
8. Conversion of solids
9. Frustum of a Cone
10. Section of a Cone
11. Conic Section
12. Parabola
13. Ellipse
14. Hyperbola
15. Identifying Conic Sections from their Equation

Mensuration Formulas For 2D Shapes

Shape	Area (Square units)	Perimeter (units)	Figure
Square	a2	4a
Rectangle	l × b	2 (l + b)
Circle	πr2	2 π r
Scalene Triangle	√[s(s−a)(s−b)(s−c)], Where, s = (a+b+c)/2	a+b+c
Isosceles Triangle	½ × b × h	2a + b
Equilateral triangle	(√3/4) × a2	3a
Right Angle Triangle	½ × b × h	b + hypotenuse + h
Rhombus	½ × d1 × d2	4 × side
Parallelogram	b × h	2(l+b)
Trapezium	½ h(a+c)	a+b+c+d

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
Cuboid	l × b × h	LSA = 2h(l + b)	2 (lb +bh +hl)
Sphere	(4/3) π r3	4 π r2	4 π r2
Hemisphere	(⅔) π r3	2 π r2	3 π r2
Cylinder	π r 2 h	2π r h	2πrh + 2πr2
Cone	(⅓) π r2 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 = 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.

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

FAQs on Mensuration

Q1: What is Mensuration?

Answer:

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?

Answer:

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?

Answer:

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

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