Mensuration - 8th math

Mensuration Class Eighth Math NCERT Exercise 11.2 Question (1) The shape of the top surface of a table is a trapezium. Find its area if its parallel sides are 1m and 1.2m and perpendicular distance between them is 0.8m.

Solution

Given, parallel sides of trapezium are 1 m and 1.2 m

Perpendicular distance between parallel sides (h) = 0.8m

Thus, Area of trapezium = ?

We know that, Area of a trapezium = 1/2 × (a + b) × h

Where a and b are parallel sides and h is the height of trapezium.

Thus, Area of given trapezium = 1/2 × (1 + 1.2) × 0.8 m²

= 1.1 × 0.8 m²

= 0.88 m²

Thus Area of given trapezium shaped table top = 0.88m² Answer

Mensuration Class Eighth Math NCERT Exercise 11.2 Question (2) The area of a trapezium is 34 cm² and the length of one of the parallel sides is 10 cm and its height is 4 cm. Find the length of the other parallel side.

Solution

Given, Area of a trapezium = 34 cm²

Height of trapezium = 4 cm

And, one parallel side of trapezium (a) = 10 cm

Thus, second parallel side of trapezium (b) = ?

We know that, Area of a trapezium = 1/2 × (a + b) × h

Where a and b are parallel sides and h is the height of trapezium.

Thus, area of given trapezium = 1/2 × (10 + b) × 4

⇒ 34 = (10 + b) × 2

⇒ (10 + b) × 2 = 34

⇒ 10 + b = 34/2

⇒ 10 + b = 17

⇒ b = 17 – 10

⇒ b = 7 cm

Thus, other parallel side of given trapezium = 7 cm Answer

Mensuration Class Eighth Math NCERT Exercise 11.2 Question (3) Length of the fence of a trapezium shaped field ABCD is 120 m. If BC = 48 m, CD = 17m and AD = 40m, find the area of this field. Side AB is perpendicular to the parallel sides AD and BC.

Solution

Given, ABCD is a trapezium.

And length of fence of this trapezium = 120 m

And, AD (a) = 40 m

And BC (b) = 48 m

CD = 17 m

And side AB is perpendicular to the parallel sides AD and BC.

Thus, Area of this trapezium = ?

We know that, length of fence = perimeter of given trapezium

And, now, perimeter of the given trapezium = BC + CD + AD + AB

⇒ 120 m = 48m + 17m + 40m + AB

⇒ 120m = 105 m + AB

⇒ 105m + AB = 120m

⇒ AB = 120 m – 105m

⇒ AB = 15m

Thus, height of the trapezium (AB) (h) = 15 m

Now, we know that, Area of a trapezium = 1/2 × (a + b) × h

Thus, area of given trapezium = 1/2 × (40 + 48) × 15 m²

= 1/2 × 88 × 15 m²

= 44 × 15 m²

= 660 m²

Thus, area of given trapezium = 660 m² Answer

Mensuration Class Eighth Math NCERT Exercise 11.2 Question (4) The diagonal of a quadrilateral shaped field is 24 m and the perpendiculars dropped on it from the remaining opposite vertices are 8 m and 13 m. Find the area of the field.

Solution

We know that area of a general quadrilateral which is split into two triangles = 1/2 (diagonal × sum of altitudes drawn on the diagonal from the other two vertices)

That means, area of such general quadrilateral = 1/2 × d(h₁ + h₂) square unit

Here, diagonal (d) = 24 m

And one altitude (h₁) = 13 m

And other altitude (h₂ = 8 m

Thus, area of given general quadrilateral

= 1/2 × 24 × (13 + 8)

= 12 × 21

= 252 m²

Thus area of given quadrilateral = 252 m² Answer

Alternate Method

This quadrilateral is made of two triangles, i.e. Δ ACD and Δ ABC

For triangle ACD

Base of the triangle (b) = 24 m

And height of the triangle (h) = 13m

Now, we know that, Area of a triangle = 1/2 × Base × Height

Thus, area of given triangle = 1/2 × 24 m × 13m

= 12 m × 13m

= 156 m²

Thus, area of Δ ACD = 156 m²

For triangle ABC

Base of the triangle (b) = 24 m

And height of the triangle (h) = 8m

Now, we know that, Area of a triangle = 1/2 × Base × Height

Thus, area of given triangle = 1/2 × 24 m × 8m

= 12 m × 8m

= 96 m²

Thus, area of Δ ACD = 96 m²

Thus, area of given quadrilateral = Area of Δ ACD + Area of Δ ABC

= 156 m² + 96 m²

= 252 m²

Thus area of given quadrilateral = 252 m² Answer

Mensuration Class Eighth Math NCERT Exercise 11.2 Question (5) The diagonals of a rhombus are 7.5 cm and 12cm. Find its area.

Solution

Given, Diagonals of a rhombus = 7.5 cm and 12 cm

Thus, area of given rhombus = ?

We know that, Area of a rhombus = 1/2 × product of diagonals

Thus, Area of given rhombus = 1/2 × 7.5 cm× 12 cm

= 7.5 cm × 6 cm

= 45.0 cm²

Thus, Area of given rhombus = 45 cm² Answer

Mensuration Class Eighth Math NCERT Exercise 11.2 Question (6) Find the area of a rhombus whose side is 5 cm and whose altitude is 4.8 cm. If one of its diagonals is 8 cm long, find the length of the other diagonal.

Solution

Given, side of rhombus = 5 cm

And Altitude of rhombus = 4.8 cm

One diagonal of the rhombus = 8 cm

Then area and other diagonal = ?

We know that, Area of a rhombus whose sides and altitude are given = side × altitude

Thus area of given rhombus = 5cm × 4.8cm

= 24 cm²

Thus, area of given rhombus = 24 cm²

Now, again we know that, Area of a rhombus whose diagonals are given = 1/2 × product of diagonals

Here, as given one diagonal (d₁) = 8 cm

And area of rhombus = 24 cm² (as calculated above)

Thus, second diagonal (d₂) = ?

Thus, Area of given rhombus = 1/2 × 8 cm × d₂

⇒ 24 = 4 cm × d₂

⇒ d₂ = 6 cm

Thus, Area of the rhombus = 24 cm² and length of other diagonal = 6 cm Answer