9th Class Mathematics Related to Competitive Exam Mensuration


Category : 9th Class




Learning Objectives


  • Introduction
  • Area of Plane Geometrical Figures
  • Quadrilateral
  • Solids




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 \[=\frac{\sqrt{3}}{4}\,{{(side)}^{2}}=\frac{\sqrt{3}}{4}{{a}^{2}}\]




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.





(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





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 \[=\frac{1}{2}\left( Sum of parallel sides \right)\times \left( distance between them \right)\]




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




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 \[=\left[ 2(l+b)\times h \right]\]




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 \[=4{{a}^{2}}\]




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 \[=(2\pi rh\,+\,2\pi {{r}^{2}})=2\pi r(h+r)\]


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)


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



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 \[=(4\pi {{r}^{2}})\]




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 \[=(3\pi {{r}^{2}})\]





Commonly Asked Questions                                                                                                                                                                                                                                                                                                                                                                                                                                     


  • 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


Answer (b)

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


Answer (a)

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


Answer: (c)

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


Answer (b)

Explanation: Let ABCD be rhombus.

\[\angle AOB\] is a right angle




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

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


  • 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


Answer (a)

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


Answer (d)

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

\[r=\frac{3}{\pi }\]

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

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

Notes - Mensuration

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