Mensuration - 8th math

Important Formula

(1) Area of a Rectangle = Length × Breadth

(2) Area of Square = Side²

(3) Area of a triangle = `1/2xx`base× height

(4) Area of a parallelogram = Base × Height

(5) Area of a circle = π × Radius²

(6) Perimeter of a Rectangle = 2(Length + Width)

(7) Perimeter of a Square = 4 × Side

(8) Perimeter of a triangle = Sum of all the three sides

(9) Circumference of a Circle = 2 π r

(10) Circumference of a Semicircle = π r

(11) Area of a Semicirlce = 1/2 × π r²

NCERT Solution Exercise 11.1

Mensuration Class Eighth Math NCERT Exercise 11.1 Question (1) A square and a rectangular field with measurements as given in the figure have the same perimeter. Which field has larger area.

Solution

Calculation for given Square

Given, side of square = 60m

Thus, area = ?

We know that, perimeter of a square = 4 × side

Thus, perimeter of the given square = 4 × 60m

= 240 m

Now, We know that, Area of a square = side²

Thus, Area of the given square = (60 m)²

= 60 m × 60 m

= 3600 m²

Calculation for given Rectangle

Given, Length of the rectangle = 80 m

And given, the perimeter of the given square = perimeter of the given rectangle

Thus, area of rectangle = ?

As calculated above the perimeter of the square = 240 m = Perimetre of the rectangle

Thus, perimeter of the rectangle = 240 m

Now, we know that, Perimeter of a rectangle = 2 (Length + Width)

Let width of the given rectangle = w

Thus, perimeter of the given rectangle = 2 (80 m + w)

⇒ 240 m = 2 (80 m + w)

⇒ 2 (80m + w) = 240 m

`=>80m +w=(240m)/2`

⇒ 80 m + w = 120 m

⇒ w = 120 m – 80 m

⇒ w = 40 m

Thus, width of the given rectangle = 40 m

Now, we know that, area of a rectangle = Length × Width

Thus, area of the given rectangle = 80 m × 40 m

= 3200 m²

Now, Area of the given square = 3600 m² and area of the given rectangle = 3200 m²

Thus, clearly, the area of the square is greater Answer

Mensuration Class Eighth Math NCERT Exercise 11.1 Question (2) Mrs Kaushik has a square plot with the measurement as shown in the figure. She wants to construct a house in the middle of the plot. A garden is developed around the house. Find the total cost of developing a garden around the house at the rate of ₹55 per m².

Solution

Given, side of square plot = 25 m

Length of rectangular house inside the plot = 20m

Width of rectangular house inside the plot = 15m

Cost of development of garden = ₹55 per m²

Thus, total cost of developing the garden outside the house = ?

Calculation of Area of Square Plot

We know that area of a square = Side²

Thus, area of given square plot = (25 m)²

= 625 m²

Calculation of Area of Rectangular house

We know that Area of a Rectangle = Length × Width

Thus, area of given rectangular house = 20 m × 15 m

= 300 m²

Calculation of Area of Garden around the house

Thus, the area of garden around the house = Area of Square Plot – Area of Rectangular House

= 625 m² – 300 m²

⇒ Area of Garden around the house = 325 m²

Calculation of Cost of Developing the Garden around the house

∵ Cost of development of 1m² of garden = ₹ 55

∴ Cost of development of 325m² of garden = ₹ 55 × 325

= ₹ 17875

Thus, cost of developing the garden around the house = ₹ 17875.00 Answer

Mensuration Class Eighth Math NCERT Exercise 11.1 Question (3) The shape of a garden is rectangular in the middle and semi circle at the ends as shown in the diagram. Find the area and the perimeter of this garden [Length of rectangle is 20 – (3.5 + 3.5) meters]

Solution

The Length of the whole garden = 20m

And, width of the garden = 7 m

Garden is rectangular in middle and semicircular at both the ends.

Thus, Area and perimeter of the garden = ?

Calculation of dimension of rectangular garden inside the plot

Because the garden is semicircular at both ends.

And, given the width of the garden = 7 m

This means diameter of the semicircle = 7 m

Thus, radius of the semicircle = Diameter/2

= 7m / 2 = 3.5 m

Thus, radius of the semicircle = 3.5 m

Thus, length of the rectangular garden in the middle of the plot = Length of the garden – (Radius of one end of semicircle + Radius of other end of the semicircle)

= 20 m – (3.5m + 3.5m)

= 20m – 7m = 13m

Thus, length of the rectangular plot inside the garden = 13 m

Calculation of the area of rectangular garden inside the plot

Thus, length of the rectangular garden in the middle of the plot = 13m

And, as given width of the rectangular garden = 7 m

Now, we know that Area of a rectangle = Length × Width

Thus, Area of the given rectangular garden = 13m × 7m

= 91 m²

Calculation of the area of the semicircular end of the garden

Given, the radius of the semicircular garden = 3.5 m

We know that, area of a semicircle `=1/2xxpi\r^2`

Thus, area of one semicircular end of garden `= 1/2xxpi (3.5m)^2`

`=1/2xx22/7xx3.5\ cmxx3.5 \m`

= 11 × 3.5 m × 0.5 m

= 19.25 m²

Thus, area of one semicircular end of garden = 19.25 m²

Calculation of circumference of semicircular end of garden

We know that, circumference of a semicircle = π r

Thus, circumference of given semicircle = π 3.5m

`=22/7xx3.5\ m`

= 22 × 0.5 m

= 11 m

Thus, circumference of one semicircular end of the garden = 11 m

Now, Area of the given diagram

= Area of rectangle + 2(Area of one semicircular end)

= 91m² + 2 × 19.25m²

= 91m² + 38.5m²

= 129.5 m²

Calculation of perimeter of the given garden having both ends of semicircular shape

Perimeter of given garden = (2 × length of rectangular part of garden) + (2 × cirmumference of one semicircular end)

= (2 × 13m) + (2 × 11m)

= 26m + 22m

= 48 m

Thus, area of the given garden = 129.5 m² and perimeter = 48m Answer

Mensuration Class Eighth Math NCERT Exercise 11.1 Question (4) A flooring tile has the shape of a parallelogram whose base is 24cm and the corresponding height is 10cm. How many such tiles are required to cover a floor of area 1080m²? (If required you can split the tiles in whatever way you want to fill up the corners).

Solution

Given, Base of tiles having parallelogram shape = 24cm

`=24/100 m`

= 0.24 m

And Height of that tiles = 10cm

`=10/100 m`

= 0.1 m

Area of floor which is to cover with given tiles = 1080m²

Thus, number of tiles to cover the given floor = ?

We know that, Area of Parallelogram = Base × Height

Thus, area of tiles having shape of parallelogram = 0.1m × 0.24m

= 0.024 m²

Thus, area of one given tile = 0.024m²

Now,

∵ To cover the area of 0.024m² number of tile required = 1

∴ To cover the area of 1m² number of tiles required = 1/0.024

∴ To cover the area of 1080m² number of tiles required `=1/(0.024)xx1080`

= 45000

Thus, required number of tiles to cover the given floor = 45000 Answer

Mensuration Class Eighth Math NCERT Exercise 11.1 Question (5) An ant is moving around a few food pieces of different shapes scattered on the floor. For which food-piece would the ant have to take a longer round? Remember, circumference of a circle can be obtained by using expression c = 2πr, where r is the radius of the circle.

Solution

Circumference or perimeter of the given shape of food peace will give the length of the path to go by the ant.

Circumference of Shape (a)

Given, The diameter of the semicircle = 2.8 cm

Thus, radius of the given semicircle = 2.8/2 cm = 1.4 cm

Now, we know that, circumference of a semicircle = π r

Thus, circumference of the given semicircle = π 1.4 cm

`=22/7xx1.4\ cm`

= 22 × 0.2 cm

= 4.4 cm

Thus, circumference of the given semicircle = 4.4 cm

Now, the perimeter of the given semicircle = circumference + diameter

= 4.4 m + 2.8 m

= 7.2 m

Thus, ant has to move 7.2 m to go round this shape (a)

Peimeter of Shape (b)

Given, Length of the rectangular shape = 2.8 cm

And, the width of the rectangular shape = 1.5 cm

And base of the semicircular shape = 2.8 cm

Thus, radius of the semicircle = 2.8 cm/2 = 1.4cm

Calculation of the perimeter of the rectangular shape

We know that, Perimeter of a rectangle = 2(Length + Width)

Thus, perimeter of the given rectangular shape = 2(2.8 cm + 1.5 cm)

= 2 × 4.3 cm

= 8.6 cm

Now, perimeter of the given rectangular shape excluding base

= perimeter of rectangle – Base

= 8.6 cm – 2.8 cm

= 5.8 cm

Thus, perimeter of the given rectangular shape without base = 5.8 cm

Calculation of the circumference of semicircular shape in the given rectangular shape

The radius of the given semicircular shape = 1.4 cm

Now, we know that circumference of a semicircle = π r

Thus, the circumference of the given semicircular shape = π 1.4cm

`=22/7xx1.4\cm`

= 22 × 0.2 cm

= 4.4 cm

Calculation of the perimeter of the given rectangular shape with a semicircular cut

Perimeter of the given shape = perimeter of the rectangular shape without base + Circumference of the semicircular cut without base

= 5.8 cm + 4.4 cm

= 10.2 cm

Thus, length has to cover by ant to go round the shape = 10.2 cm

Peimeter of Shape (c)

Given, The diameter of the semicircle = 2.8 cm

Thus, radius of the given semicircle = 2.8/2 cm = 1.4 cm

Now, we know that, perimeter of a semicircle = π r

Thus, perimeter of the given semicircle = π 1.4 cm

`=22/7xx1.4\ cm`

= 22 × 0.2 cm

= 4.4 cm

Length of each of the two sides of the conical shape = 2 cm

Thus, perimeter of the given semicircular and conical shape = circumference of the semicircle + sum of two sides of the cone

= 4.4 cm + 4 cm

= 8.4 cm

Thus, ant has to cover a distance of 8.4 cm to go round this shape (c)

Now, the perimeter of the shape (a) = 7.2 cm

And, perimeter of the shape (b) = 10.2 cm

And, perimeter of the shape (c) = 8.4 cm

Thus, clearly ant has to cover longest round to go round the food piece of shape (b) i.e. equal to 10.2 cm Answer

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