# 6th Class Mental Ability Mensuration

Mensuration

Category : 6th Class

Mensuration

Learning Objectives

• Mensuration
• Perimeter of Geometrical Shapes
• Area of Geometrical Shapes

Mensuration

Mensuration is the branch of mathematics which deals with the measurement of lengths, area and volume of the plane and solid figures.

Perimeter of a plane figure: The distance all round a plane figure is called perimeter of the figure or the lengths of boundary of a plane figure is known as its perimeter

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Perimeter of the quadrilateral $ABCD=AB+BC+CD+DA$

$=40\text{ }cm+10\text{ }cm+40\text{ }cm+10\text{ }cm$

$=100\text{ }cm$

Perimeter of the hexagon $ABCDEF=AB+BC+CD+DE+EF+FA$

$=100\text{ }m+120\text{ }m+90\text{ }m+45\text{ }m+60\text{ }m+80\text{ }m$

$=495\text{ }m$

Perimeter of a scalene triangle = Sum of all the three sides of the triangle.

Example: Find the perimeter of the triangle ABC.

Perimeter of the triangle $ABC=AB+BC+CA$

$=4\text{ }cm+12\text{ }cm+8\text{ }cm$

$=24cm$

Perimeter of a rectangle $=2\times \left( length+breadth \right).$

Example: Find the perimeter of the following rectangle ABCD.

Perimeter of rectangle $ABCD=2\times \left( AB+BC \right)$

$=2\times \left( 15cm+9cm \right)$

$=48\text{ }cm$

Perimeter of regular shapes

Perimeter of an equilateral triangle $\text{=3 }\!\!\times\!\!\text{ length of one side}$.

Example: Find the perimeter of the given triangle.

Perimeter of the triangle $=3\times 4\text{ }cm$

$=12\text{ }cm$

Perimeter of a square: $\text{4 }\!\!\times\!\!\text{ length of one side}\text{.}$

Example: Find the perimeter of the given square.

Perimeter of the square $=4\times 1\text{ }m$

$=4\text{ }m$

Perimeter of regular pentagon $=5\text{ }\times \text{ }length\text{ }of\text{ }one\text{ }side.$

Example: Find the perimeter of the given pentagon.

Perimeter of the pentagon $=5\times 4\text{ }cm$

$=20\text{ }cm$

Perimeter of the regular hexagon $=6\times length\text{ }of\text{ }one\text{ }side.$

Perimeter of the hexagon $=6\times 5\text{ }cm$

$=30\text{ }cm.$

Area of a plane figure: The measurement of the region enclosed by a plane figure is called area of the figure or area is the amount of surface covered by the shape.

Area of triangle $=\frac{1}{2}\times base\times height.$

Example: Find the area of the given triangle.

Area of the triangle $=\frac{1}{2}\times 4cm\times 6cm=12c{{m}^{2}}$

Area of rectangle: $length\times breadth.$

Example: Find the area of the given rectangle.

Area of the rectangle $=8\text{ }cm\times 6\text{ }cm=48\text{ }c{{m}^{2}}$

Area of square = $side\times side$

Example: Find the area of the given square.

Area of the square $=6cm\times 6cm=36c{{m}^{2}}.$

Find the area of a square park whose perimeter is 320 m.

(a) $6300{{m}^{2}}$ (b) $6500{{m}^{2}}$

(c) $6400{{m}^{2}}$  (d) $6200{{m}^{2}}$

(e) None of these

Explanation: Let the length of each side of the square park be a metre. Then, perimeter = 320 m

$\Rightarrow 4a=320$

$\Rightarrow a=\frac{320}{4}=80m$ [$\therefore$ Perimeter of a square$=4\times Side$]

$\therefore Area={{a}^{2}}=\left( 80\times 80 \right){{m}^{2}}=6400{{m}^{2}}.$

Find the breadth of a rectangular plot of land, if its area is 440 sq. m and length is 22 m. Also, find its perimeter.

(a) 20 m, 84 m               (b) 2 m, 46 m

(c) 50 m, 42 m.              (d) 4 m, 40 m

(e) None of these

Explanation: We have,

I = Length of the plot = 22 m. Area of the plot = 440 sq. metre

Let the breadth of the plot be b metres. Then,

$Breath=\frac{Area}{Length}\Rightarrow b=\frac{420}{22}=20m$

Perimeter $=2\left( I+b \right)=2\left( 22+20 \right)m=2\times 42m=84m.$

Hence, the breadth of the plot is 20 m and the perimeter is 84 m.

The carpet for a room 6.6 m by 5.6 m costs Rs. 3960 and it was made from a roll 70 cm wide. Find the cost of the carpet per metre.

(a) Rs. 70                       (b) Rs. 78

(c) Rs. 75                      (d) Rs. 70

(e) None of these

Explanation: We have,

Area of the carpet $=6.6\times 5.6=36.96{{m}^{2}},$ Width of the roll = 70 cm = 0.7 m

$\therefore$Length of the roll $=\frac{Area}{With}=\frac{36.96}{0.7}m=52.8m$

Cost of the carpet $=Rs.\,3960$

$\therefore$Cost of the carpet per metre $=Rs.\,\frac{3960}{52.8}=Rs.\,75$

Hence, the carpet costs $Rs.\,75$ per metre.

A rectangular lawn of length 40 m and breadth 25 m is to be surrounded all around by a path which is 2 m wide. Find the area of the path.

(a) $276\text{ }{{m}^{2}}$

(b) $345\text{ }{{m}^{2}}$

(c) $308\text{ }{{m}^{2}}$

(d) All of these

(e) None of these

Explanation: Area of the path $=2\left( 44\times 2 \right)+2\left( 25\times 2 \right)=176+100=276{{m}^{2}}.$

A garden is 60 m by 30 m. It has two paths at the centre as shown in the figure. If the width of the path is 2 m, how much area is left for gardening?

(a) $1441\text{ }{{m}^{2}}$

(b) $1624\,{{m}^{2}}$

(c) $1328\,{{m}^{2}}$

(d) All of these

(e) None of these

Explanation: Area for gardening $=4\left( 14\times 29 \right)=1624{{m}^{2}}.$