Mensuration Class 8 Notes- Chapter 11

What is Mensuration?

Mensuration is a branch of mathematics which mainly deals with the study of different kinds of Geometrical shapes along with its area, length, volume and perimeters. It is completely based on the application of both algebraic equations and geometric calculations. The results obtained by the Mensuration are considered very accurate. There are two types of geometric shapes:

Volume of Solids

Volume of a 3D Object

Volume is the space occupied by the three dimensional object. It is a three dimensional quantity.

Volume of a Cuboid

Mensuration 01

Volume of a cuboid =l×b×h

where, l is the length, b is the breadth and h is the height of the cuboid.

Volume of a Cube

Mensuration 02

Volume of a cube =l3

Where, l is the length of the each side of the cube.

Volume of a Cylinder

Mensuration 03

Volume of the cylinder =πr2h

Where r is the radius of the base and h is the height of the cylinder.

Basics Revisited

Introduction to Mensuration

Mensuration is the study of geometry that deals with the measurement of length, areas and volumes.

Perimeter is the total length or path of a given shape.
Area is the total region covered by the given shape.
Volume is the total space occupied by the given shape.

Identifying Shapes and Areas of Different Regular Figures

Area of a Rectangle : length×breadth,  perimeter : 2(length+breadth)

Area of Square : side×side, perimeter : 4×side

Area of  Triangle : 1/2 (base×height), perimeter : a+b+c (sum of 3 sides)

Area of Parallelogram : base×height , perimeter : 2(length+breadth)

Area of Circle : π×(radius)2, perimeter : 2×π×radius.

Trapezium

Area of Trapezium by Division into Shapes of Known Area

Consider the trapezium where a and b are parallel sides, h is the height. Trapezium is divided into 3 parts : two triangles, one rectangle.
Mensuration 04
Here h is the height, a and b are 2 parallel sides.

Area of trapezium = Area of 2 triangles + Area of rectangle
=[\(\frac{1}{2}\)×c×h+\(\frac{1}{2}\)×d×h]+[a×h]
=[\(\frac{1}{2}\)×c×h+\(\frac{1}{2}\)×d×h]+[\(\frac{1}{2}\) ×2a×h]  [Multiplying and dividing by 2] =\(\frac{1}{2}\)×h(c+d+2a)
=\(\frac{1}{2}\)×h(c+d+a+a)
=\(\frac{1}{2}\)×h(b+a)    [ c + d + a = b]

Area of trapezium =\(\frac{1}{2}\)×h (sum of 2 parallel sides)

Area of Trapezium by Finding the Area of a Triangle of Same Area

The area of the trapezium can be found out by dividing it into a triangle and a polygon.

Consider a trapezium WXYZ. Mark a midpoint A for side XY and join AZ. Cut the trapezium along AZ and obtain a ΔAZY
Flip the ΔAZY and place it as shown below. Now the new polygon is a triangle.
Mensuration 05
We know that,
Area of a triangle=\(\frac{1}{2}\)×base×height
Substituting the values we get,
Area of a triangle=\(\frac{1}{2}\)×(a+b)×h
But the original polygon is a trapezium. So,

Area of a trapezium=\(\frac{1}{2}\)×(a+b)×h

Quadrilaterals in General

Area of a General Quadrilateral

Consider a quadrilateral ABCD. Draw diagonal AC. From B and D draw perpendiculars h1,h2 to AC
Mensuration 06
Area of quadrilateral = Area of triangle ABC + Area of triangle ADC
=\(\frac{1}{2}\)×base×height+12×base×height
=(\(\frac{1}{2}\)×AC×h1)+(\(\frac{1}{2}\)×AC×h2)  [Where, h1, h2 are the heights, AC is the base] =\(\frac{1}{2}\)×AC×(h1+h2)
=\(\frac{1}{2}\)×d×(h1+h2)  [ AC is a diagonal]

Area of a Quadrilateral=\(\frac{1}{2}\)×d×(h1+h2)

where d is diagonal and h1,h2 are perpendicular drwan to a diagonal.

Area of Rhombus

Area of rhombus =\(\frac{1}{2}\)×d1×d2,
Mensuration 07
where d1 and d2 are the diagonals.

Area of Polygons

The area of any given polygon can be found by cutting the polygon into shapes whose area is known and adding the area of these shapes.

Some of the ways to find the area is shown below.
Mensuration 08
Area of this polygon = area of 2 trapeziums
Mensuration 09
Area of this polygon = Area of 2 triangles + Area of rectangle.
Mensuration 10
Area of this polygon = Area of 4 triangles.

Surface Area of Solids

Solid Shapes

Solid shapes or solid figures are the three dimensional figures which have length, breadth and height. Using these, surface areas and volumes of these figures are found out.

Solids with a Pair or More of Identical Faces

Solids with a pair of identical faces are:
Mensuration 11

Surface Area of Solid Shapes

The surface area of the object is the total area occupied by the surface of the object.
or  surface area is simply the sum of the areas of the flat surfaces (called faces).

Surface Area of a Cuboid

Mensuration 12

Total Surface area of cuboid =2(lb+bh+lh)

Lateral Surface area of cuboid =2h(l+b)

Where, l is the length, b is the breadth and h is the height.

Surface Area of a Cube

Mensuration 13

Total Surface area of a cube =6l2

Lateral Surface area of a cube  =4l2

Where l is the length of each side of the cube.

Surface Area of a Cylinder

Mensuration 14

Curved surface area of cylinder (C.S.A) =2πrh

Total Surface area of cylinder(T.S.A) =2πr(r+h)

Where, r is the radius of the cylinder and h is the height of the cylinder.

Summary of Mensuration

Relation between Volume and Capacity

Volume is the total space occupied by an object. Volume is measured in cubic units,
Capacity refers to the maximum measure of an object’s ability to hold a substance, like a solid, a liquid or a gas. Capacity can be measured in almost every other unit, including liters, gallons, pounds, etc.
Eg : A bucket contains 9 litres of water, then its capacity is 9 litres.

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