Are you thinking of skipping Mensuration?
The Mensuration unit of ICSE Class 9 discusses how to measure the area and perimeter of plane figures. Read this article to easily learn how to measure geometrical shapes.
By Topperlearning Expert 19th Aug, 2020 | 02:02 pm
ShareMost 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.
Area of a triangle
Area of a triangle using Heron’s formula = ,
where a, b and c are the three sides of a triangle and s is a semi-perimeter which is equal to .
- Some Special Types of Triangles
- Equilateral Triangle
- Equilateral Triangle
Area of an equilateral triangle =
- Isosceles Triangle
Area of an isosceles triangle =
where a = length of each equal side
and b = length of the base
- Right-angled Triangle
Area of a right-angled triangle
=
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
- Rectangle
i. Area = length × breadth = l × b
ii. Perimeter = 2(length + breadth) = 2(l + b)
iii. Length of a diagonal (d) =
- Square
i. Area = (side)2 = a2
ii. Perimeter = 4 × side = 4a
iii. Length of a diagonal =
- Parallelogram
Area = base × height
- Rhombus
i. Perimeter = 4 × side = 4a
ii. Area =
- Trapezium
Area of a trapezium
= × (sum of parallel sides) × (distance between parallel sides)
= (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
- Volume = length × breadth × height = l × b × h
- Total surface area = 2(l × b + b × h + h × l)
- Lateral surface area of a cuboid
= Sum of the areas of the four walls (vertical faces) of the cuboid
= 2(l + b) × h - Length of a diagonal =
5. Cube
- Volume = a3 = (edge)3
- Total surface area = 6a2
- Lateral surface area = 4a2
- Length of a diagonal =
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
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:
- 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.
- 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.
- 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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