Notes - Mensuration

**Category : **11th Class

**Learning Objectives**

- Introduction
- Area of Plane Geometrical Figure
- Quadrilateral
- 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°.
- 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 = \[\frac{1}{2}~\left( perpendicular \right)\times Base=\frac{1}{2}\times AB\times BC\]

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 of isosceles triangle \[=\frac{1}{2}\times BC\times AD=\frac{1}{4}b\sqrt{4{{a}^{2}}-{{b}^{2}}}\]

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° 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° is given by\[L=\frac{n}{360}\times \pi {{r}^{2}}.\]

Quadrilateral 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 = \[(I\times b)\]square units

(ii) Length = \[\frac{area}{breadth},breadth=\frac{area}{length}\]

(iii) \[Diagonal\,=\sqrt{{{I}^{2}}+{{b}^{2}}}\,\,\,units\]

(iv) Perimetre = 2(I + 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 - \[\left\{ \frac{1}{2}\times ()diagonal{{)}^{2}} \right\}sq\,\,units\]

(iii) Diagonal = \[a\sqrt{2}\]units

(iv) Perimetre = 4a units

- Area of parallelogram = \[base\times height\]
- Area of a Rhombus = \[\frac{1}{2}\times \left( product\text{ }of\text{ }diagonals \right)\]
- Area of a Trapezium = \[\frac{1}{2}\left( Sum\text{ }of\text{ }parallel\text{ }sides \right)\times (distance\text{ }between\text{ }them)\]

**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 =\[(I\times b\times h)\]

(ii) Total surface area = 2 \[(Ib\times bh\times \text{I}h)\]

(iii) Lateral surface area = \[\left[ 2\left( I+b \right)\times h \right]\]

**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 = \[{{a}^{3}}\]

(jj) 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 surf ace 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 \[I=\sqrt{{{h}^{2}}+{{r}^{2}}}\]then we have

(i) Volume of the cone = \[\frac{1}{3}\pi {{r}^{2}}h\]

(ii) Curved surface area of the cone = \[\pi rl\]

(iii) Total surface area of the cone = (curved surface area) + (area of the base)

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

**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 equai parts/ each part is called hemisphere. For a hemisphere of radius r, we have:

(i) Volume of the hemisphere = \[\frac{2}{3}\pi {{r}^{3}}\]

(ii) Curved 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) 7.5 sq. m

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

(e) None of these

**Ans. (b)**

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

\[\therefore \] Area =\[\frac{1}{2}\times 6\times 2.5=7.5\,\,sq.\,\,m.\]

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

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

(c) 21,000 sq. m. (d) 22.000 sq. m.

(e) None of these

**Ans. (a)**

**Explanation:** Let length and breadth (in metres) respectively be 5*X* and 3*x *2(5*x*+3*x*) = 640

\[16x=640\Rightarrow X=40\]

\[\therefore \] Area = \[5X\times 3X=15{{x}^{2}}=15{{\left( 40 \right)}^{2}}=24,000\text{ }sq.\text{ }m.\]

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

(a) 110.5 sq. m (b) 111.4 sq. m

(c) 112.5 sq. m (d) 110.3 sq. m

(e) None of these

**Ans. (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

**Ans. (b)**

**Explanation:** Let ABCD be rhombus.

\[\angle AOB\]is a right angle

\[OB=\frac{1}{2}\left( BD \right)=6\text{ }cm,\text{ }AB=10\text{ }cm.\]

\[\therefore \,\,\,OA=\sqrt{100-36}=8cm\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\Rightarrow \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,AC=2(AO)=16\,cm.\]

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

**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

**Ans. (a)**

**Explanation:** If r be the radius,

\[2\pi r-2r=210\]

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

**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

**Ans. (d)**

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

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

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

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

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