9th Class Mathematics Related to Competitive Exam Mensuration

Mensuration

Category : 9th Class

MENSURATION

Learning Objectives

• Introduction
• Area of Plane Geometrical Figures
• Solids

Introduction

Mensuration is a science of measurement of the lengths of lines, area of surfaces and volumes of solids.

Some Important Definitions and Formulae:

• If any closed figure has three sides then it is called a triangle.
• In a triangle the sum of three angles is $180{}^\circ$.
• In a triangle the sum of the lengths of any two sides should be more than the third side.
• Similarly the difference between any two sides of a triangle is less than the third side.
• The side on which a triangle rests is called the base. The length of the perpendicular drawn on the base from opposite vertex is called the height of the triangle.
• If the three sides of a triangle have three different lengths then it is called a scalene triangle.
• If exactly two side of a triangle are equal and the third side has different length then it is called an isosceles triangle.
• If all the three sides of a triangle are equal then it is called an equilateral triangle.

Area of Plane Geometrical Figures

• Triangle

(i) Right Triangle

(ii) Scalene Triangle (Heron's formula)

(iii) Isosceles Triangle

(iv) Equilateral Triangle

Right Triangle: Area of right triangle $=\frac{1}{2}\times \left( perpendicular \right)\times Base=\frac{1}{2}\times AB\,\times BC$

Scalene Triangle (Heron's formula): Let, a, b, c be the length of sides of a triangle

then area $=\sqrt{s(s-a)(s-b)(s-c)}$ sq. unit, where $s=\frac{1}{2}(a+b+c)$

Isosceles Triangle: Area ot isosceies trsang Se

$=\frac{1}{2}\times BC\times AD=\frac{1b}{4}\sqrt{4{{a}^{2}}-{{b}^{2}}}$

Equilateral triangle:

Area $=\frac{\sqrt{3}}{4}\,{{(side)}^{2}}=\frac{\sqrt{3}}{4}{{a}^{2}}$

Circle

A circle is a geometrical figure consisting of all those points in a plane which are at a given distance from a fixed point in the same plane. The fixed point is called the centre and the constant distance is known as the radius.

A circle with centre O and radius r is generally denoted by C (0, r).

Circle Formulas

1. The circumference C of a circle of radius r is given by the formula $C=2{}^\circ \,\pi r$.
2. The area A of a circle of radius r is given by the formula $A=\pi {{r}^{2}}$.
3. The areas of two circles are to each other as the squares of their radii.
4. The length L of an arc of $n{}^\circ$ in a circle of radius r is given by the formula $L=\frac{n}{360}\times 2\pi r$
5. The area A of a sector of a circle of radius r with central angle of $n{}^\circ$ is given by $A=\frac{n}{360}\times \pi {{r}^{2}}$

We know that a geometrical figure bounded by four lines segment is called quadrilateral In this section we will study about area and perimetre of different quadrilaterals.

Rectangle

 (i) Area $=(l\,\times b)$square units (ii) Length $=\frac{area}{breadth},$ breadth $=\frac{area}{length},$ (iii) $\operatorname{Diagonal}\,=\sqrt{{{l}^{2}}+{{b}^{2}}}\,units$ (iv) Perimetre $=2(l+b)$units

Square

Let ABCD be a square with each side equal to 'a' units then we have

 (i) Area $={{a}^{2}}$ sq units (ii) Area $=\left\{ \frac{1}{2}\times {{(diagonal)}^{2}} \right\}$ units (iii) Diagonal $=a\sqrt{2}$ units (iv) Perimetre = 4a units

• Area of parallelogram $~= base\,\times height$
• Area of a Rhombus $=\frac{1}{2}\times (product\,\,of\,diagonals\,)$
• Area of a Trapezium $=\frac{1}{2}\left( Sum of parallel sides \right)\times \left( distance between them \right)$

Solids

The objects having definite shape and size are called solids. A solid occupies a definite space.

Cuboid

Solids like matchbox, chalk box, a tile, a book an almirah, a room etc. are in the shape of a cuboid.

.

For cuboid of length = I, breath = b and height = h, we have:

 (i) Volume $=\left( I\,\times b\,\times h \right)$ (ii) Total surface area $=2 \left( Ib\,\times bh\,\times Ih \right)$ (iii) Lateral surface area $=\left[ 2(l+b)\times h \right]$

Cube

Solids like ice cubes, sugar cubes, dice etc. are the shape of cube Formula for a cube having each edge = a units, we have:

 (i) Volume $={{a}^{3}}$ (ii) Total surface area $=6{{a}^{2}}$ (iii) Lateral surface area $=4{{a}^{2}}$

Cylinder

Solids like measuring jar, circular pencils, circular pipes, road rollers, gas cylinders, are said to have a cylindrical shape. Formula for a cylinder of base radius = r & height (or length) = h, we have

 (i) Volume $=\pi {{r}^{2}}h$ (ii) Curved surface area$=2\pi rh$ (iii) Total surface area $=(2\pi rh\,+\,2\pi {{r}^{2}})=2\pi r(h+r)$

Cone

Consider a cone in which base radius = r, height = h & slant height $\operatorname{l}=\sqrt{{{h}^{2}}+{{r}^{2}}}$ then we have

 (i) Volume of the cone $=\frac{1}{3}\pi {{r}^{2}}h$ (ii) Curved surface area of the cone = (curved surface area) + (area of the base)

$=\pi rl+\pi {{r}^{2}}=\pi r(l+r)$

Sphere

Objects like a football, a cricket ball, etc. are said to have the shape of a sphere. For a sphere of radius r, we have

 (i) Volume of the sphere $=\left( \frac{4}{3}\pi {{r}^{3}} \right)$ (ii) Surface area of the sphere $=(4\pi {{r}^{2}})$

Hemisphere

A plane through the centre of a sphere cuts it into two equal parts, each part is called hemisphere.

For a hemisphere of radius r, we have:

 (i) Volume of the hemisphere $=\frac{2}{3}\pi {{r}^{2}}$ (ii) Total surface area of the hemisphere $=(2\pi {{r}^{2}})$ (iii) Total surface area of the hemisphere $=(3\pi {{r}^{2}})$

• Find the area of a right - angle triangle with base 6 metres and hypotenuse 6»5 metres,,

(a) 7.1 sq. m.                 (b) 75 sq. m.

(c) 7.05 sq. m.                            (d) 7.9 sq.m.

(e) None of these

Explanation: Height $=\sqrt{{{(6.5)}^{2}}-{{6}^{2}}}=\sqrt{(42.25-36)}=\sqrt{6.25}=2.5m$

$\therefore \,\,\,\,Area\,=\frac{1}{2}\times 6\times 2.5=7.2\,sq\,.m.$

• The perimeter of a rectangle is 640 meters and the length is to the breadth as 5 : 3, find its area.

(a) 24,000 sq. m.            (b) 23,000 sq. m.

(c) 21,000 sq. m.                        (d) 22,000 sq. m.

(e) None of these

Explanation: Let length and breadth (in metres) respectively be 5X and 3x

$2\left( 5x+3x \right)=640$

$16x=640\,\Rightarrow 40$

$\therefore \,\,\,Area\,=\,5X\times 3X=15{{x}^{2}}=15{{(40)}^{2}}=24,000\,sq.\,m.\,$

• What is the area of a square whose diagonal is 15 metres?

(a) 110. 5sq. m               (b) 111. 4sq. m

(c) 112. 5sq. m               (d) 110. 3sq. m

(e) None of these

Explanation: Area of sq. $=\,\frac{1}{2}\times {{(15)}^{2}}=112.5\,sq.m$

• Find the area of a rhombus one side of which measures 10 cm and one diagonal 12 cm

(a) 95 sq. m                   (b) 96 sq. m

(c) 92 sq. m                   (d) 94 sq. m

(e) None of these

Explanation: Let ABCD be rhombus.

$\angle AOB$ is a right angle

$\operatorname{OB}=\frac{1}{2}(BD)\,=6cm,\,\,AB=10\,cm.$

$\Rightarrow AO 2 \left( AO \right)=16\,cm.$

$\therefore \,\,\,Area of rhombus =\frac{1}{2}\left( AC\times BD \right)=\frac{1}{2}\,\,\left( 16\times 12 \right)=96 sq. m.$

• The difference between the circumference and the diameter of a circle is 210 metres.

Find the radius of the circle.

(a) 49 m                        (b) 47 m

(c) 46 m                        (d) 48 m

(e) None of these

Explanation: If r be the radius,

$2\pi r-2r=210$

$r=\frac{210}{2(\pi -1)}=\frac{105}{\frac{22}{7}-1}=\frac{7\times 105}{15}=49m.$

• The circumference of the base of a cylinder is 6 metres and its height is 44 metres. Find the volume.

(a) 124 cub. m               (b) 123 cub. m

(c) 121 cub. m                (d) 126 cub. m

(e) None of these

Explanation: If r be the radius of the base, $2\pi r=6$

$r=\frac{3}{\pi }$

$\therefore \,\,\,Area\,\,of\,base\,=\pi {{r}^{2}}=p{{\left( \frac{3}{p} \right)}^{2}}=\frac{9}{\pi }\,sq.m.$

$\therefore \,\,\,Volume=\frac{9}{\pi }\,\times 44=\frac{9}{22}\times 7\times 44=126\,cub.m.$

Other Topics

Notes - Mensuration

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