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Are you thinking of skipping Mensuration?

Most students feel that the chapter Mensuration 1 and Mensuration 2 is difficult compared to other chapters as it deals with figures, and so, they try to skip this chapter (which is not correct). But you should not think of skipping Mensuration as it is the most interesting and scoring unit. Instead, you have to understand the features of all the figures, by-heart some of the basic formulae to derive more formulae and practise different kinds of problems regularly.

What is Mensuration?

Mensuration is a branch of Mathematics which deals with the measurement of the length, perimeter, circumference, area, surface area and volume of geometrical shapes in 2D and 3D space.

The Mensuration unit of ICSE Class 9 deals with the concepts of area and perimeter of plane figures. We discuss some of the shapes and their corresponding measurements in this article.

 

1.Triangles

  

Area of a triangle   equals 1 half cross times B a s e cross times H e i g h t equals 1 half cross times B C cross times A D

Area of a triangle using Heron’s formula = square root of s open parentheses s minus a close parentheses open parentheses s minus b close parentheses open parentheses s minus c close parentheses end root,

where a, b and c are the three sides of a triangle and s is a semi-perimeter which is equal to fraction numerator a plus b plus c over denominator 2 end fraction .

  • Some Special Types of Triangles
    • Equilateral Triangle

 

Area of an equilateral triangle =fraction numerator square root of 3 over denominator 4 end fraction cross times left parenthesis s i d e right parenthesis squared equals fraction numerator square root of 3 end root over denominator 4 end fraction cross times left parenthesis a right parenthesis squared

  • Isosceles Triangle

 

Area of an isosceles triangle =  1 fourth cross times b cross times square root of 4 a squared minus b squared end root

where a = length of each equal side

and b = length of the base

  • Right-angled Triangle

 

Area of a right-angled triangle

=1 half cross times P e r p e n d i c u l a r cross times B a s e equals 1 half cross times A B cross times B C

2. Quadrilaterals

             i.        When one diagonal and perpendiculars to this diagonal from the remaining vertices are given.

 

 

           ii.          When two diagonals of a quadrilateral cut each other at right angles.

 

 

  • Some Special Types of Quadrilaterals
    • Rectangle

                                 i.        Area = length × breadth = l × b

                                ii.        Perimeter = 2(length + breadth) = 2(l + b)

                              iii.        Length of a diagonal (d) =square root of 1 squared plus b squared end root

  • Square

              

                               i.          Area = (side)2 = a2

                              ii.          Perimeter = 4 × side = 4a

                            iii.          Length of a diagonal = square root of 2 a end root space o r space square root of 2 cross times A r e a end root

 

  • Parallelogram

Area = base × height

  • Rhombus

                               i.          Perimeter = 4 × side = 4a

                              ii.          Area =1 half left parenthesis P r o d u c t s space o f space d i a g o n a l s right parenthesis equals 1 half cross times B D cross times A C

  • Trapezium

Area of a trapezium

= 1 half× (sum of parallel sides) × (distance between parallel sides)

= 1 half(a + b) × h

3. Circle

 

 

Remember:

Area: The area of a plane figure is the measure of the surface enclosed by its boundary.

Units: square centimetre (cm2), square metre (m2)

 

Perimeter: The perimeter of a plane figure is the length of its boundary.

Units: cm, m

 

Circumference: The distance around the edge of a circle. It is a type of perimeter.

Units: cm, m

 

Concepts of solids (surface area and volume of 3-D solids) 

 4. Cuboid

 

  1. Volume  = length × breadth × height = l × b × h
  2. Total surface area = 2(l × b + b × h + h × l)
  3. Lateral surface area of a cuboid
    = Sum of the areas of the four walls (vertical faces) of the cuboid
    = 2(l + b) × h
  4. Length of a diagonal =square root of 1 squared plus b squared plus h squared end root

5. Cube


 

  1. Volume = a3 = (edge)3  
  2. Total surface area = 6a2 
  3. Lateral surface area = 4a2 
  4. Length of a diagonal =  square root of 3 a

Remember:

Solid: Anything which occupies space and has a definite shape is called a solid.
A solid has three dimensions—length, breadth and height.

Surface Area: The sum of the areas of all the surfaces of a solid is called its surface area or its total surface area.

Units: square centimetre (cm2), square metre (m2)

Volume: The space occupied by a solid is called its volume.

  • Capacity of a container = Its internal volume
  • Volume of material in a hollow body = Its external volume – Its internal volume

Units: cm3, m3

 

Cross-section

A cutting, or a piece of something cut off, at right angles to an axis is called a cross-section.

 

Uniform Cross-Section:

A solid is said to have a uniform cross-section if a perpendicular cuts off the same shape and size at each point of its length or height.

Examples:

  1. When a cuboid is cut through points A and B perpendicular to its length, the faces obtained (as the cross-section) at both points are of the same shape and size.


  2. When a right-circular cylinder is cut through points A and B perpendicular to its length, the faces obtained (as the cross-section) at both points are of the same shape and size.


  3. The cross-section of a right-circular cone is not uniform, since the cuts made give faces of the same shape but they are not of the same size.


 

Note that when a body has a uniform cross-section, you can use these formulae:

1. Volume = Area of cross-section × length

2. Surface area (excluding cross-section) = Perimeter of cross-section × length

 

All these concepts and formulae will surely help you to understand the chapter Mensuration, and we hope you’ll study the chapter and remember these formulae as they will also be helpful in solving the problems in your Maths exam.

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