Important Questions on Area for Class 9 | Mensuration

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Important Questions on Area

Class 9 Mensuration | Step-by-Step Solutions

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📚 Key Formulas Sheet

1. Heron’s Formula (Triangle):

Area = √ s(s – a)(s – b)(s – c)

where s = a + b + c 2

2. Trapezium Area:

Area = 1 2 h(p₁ + p₂)

3. Four Walls Area:

Area = 2h(l + b)

Excluding doors/windows = 2h(l + b) – Area of doors – Area of windows

4. Unit Conversions:

• 1 Ropani = 508.72 m²

• 1 Aana = 31.8 m²

• 1 Aana = 4 Paisa

• 1 Dhur = 182.25 ft²

Question 1

Triangular Shape Land Area & Valuation

Triangle

1. Based on the given triangular shape of land in the figure, answer the following questions:

(a) Find the area of the triangular shape of land as given in the figure.
(b) Convert the area of the land into Aana and Paisa.
(c) If per Aana of the land costs Rs 4,75,000, calculate the cost of the land.

Figure: Triangle PQR

(a) Area of Triangular Land:

Given sides: a = 55 m, b = 32 m, c = 75 m

Semi-perimeter (s):

s = a + b + c 2

or, s = 55 + 32 + 75 2

or, s = 162 2

or, s = 81 m

By Heron’s Formula:

Area = √ s(s – a)(s – b)(s – c)

or, Area = √ 81 × (81 – 55) × (81 – 32) × (81 – 75)

or, Area = √ 81 × 26 × 49 × 6

or, Area = √ 619,164

or, Area ≈ 786.87 m²

➔ Area of Land ≈ 786.87 m²

(b) Convert Area into Aana and Paisa:

We know, 1 Aana = 31.8 m²

Total Aana = 786.87 / 31.8 ≈ 24.7443 Aana

or, Whole Aana = 24 Aana

or, Remaining Aana = 0.7443 Aana

Since 1 Aana = 4 Paisa:

Paisa = 0.7443 × 4 ≈ 2.98 Paisa

➔ Area = 24 Aana 2.98 Paisa

Rate per Aana = Rs 4,75,000

Total Cost = 24.7443 × Rs 4,75,000

➔ Total Price ≈ Rs 1,17,53,542.50

Question 2

Triangular Land Ratio & Perimeter Area Calculations

Ratio

2. A triangular piece of land has perimeter 160 feet. If the sides of the land are in the ratio 5:7:8, find the area of the land in:

(a) ft²

(b) m²

Question 3

Trapezium Land Area & Unit Conversions

Trapezium

3. The figure PQRS represents a piece of land in the shape of a trapezium.

(a) Find the area of the land.
(b) Convert the area of the land in Ropani.
(c) Convert the area of the land in Kattha.
(d) If the current rate of the land is Rs 2,50,000 per Ropani, find the current price of the land.

Figure: Trapezium PQRS

⭐ Sample questions according to CDC Grid Curriculum

Question 4 – CDC Grid

Triangular Garden Sides & Fencing Cost

CDC Grid

4. The sides of a triangular garden are in the ratio 13 : 14 : 15. The area of the garden is 336 m². Answer the following questions:

(a) Write the formula to find the area of the triangle in terms of its sides a, b and c. [1]
(b) Find the length of each side of the garden. [2]
(c) How much will it cost for fencing the garden at the rate of Rs 325 per meter? [2]

(a) Formula for Area of Triangle:

Area = √ s(s – a)(s – b)(s – c)

where s = a + b + c 2

(b) Length of each side (Step-by-step vertical solution):

Let the sides be: a = 13x, b = 14x, c = 15x

Semi-perimeter (s) = 13x + 14x + 15x 2

or, s = 42x 2

or, s = 21x

Applying Area Equation:

Given Area = 336 m²

√ s(s – a)(s – b)(s – c) = 336

or, √ 21x × (21x – 13x) × (21x – 14x) × (21x – 15x) = 336

or, √ 21x × 8x × 7x × 6x = 336

or, √ 7056 x⁴ = 336

or, 84 x² = 336

or, x² = 336 84

or, x² = 4

or, x = √4 = 2

Calculating Side Lengths:

a = 13 × 2 = 26 m

b = 14 × 2 = 28 m

c = 15 × 2 = 30 m

➔ Side lengths: a = 26 m, b = 28 m, c = 30 m

Perimeter = a + b + c = 26 + 28 + 30 = 84 m

Rate = Rs 325 / m

Total Cost = Perimeter × Rate

or, Total Cost = 84 × 325

➔ Total Cost = Rs 27,300

Question 5

Room 4 Walls Area, Plastering Cost & Floor Area

4 Walls

5. The length, breadth and height of a room are 6.36 m, 5 m and 3 m respectively. If the room contains a door of size 0.75 m × 2 m and a window of size 1.5 m × 2 m:

(a) Write the formula for finding the area of four walls of the room. [1]
(b) Find the area of four walls including door and window of the room. [1]
(c) Find the total cost of plastering the four walls of the room at the rate of Rs 500 per m². [2]
(d) Find the area of the floor of the room in paise. (1 Aana = 31.8 m² = 4 paise) [1]

Question 6

Triangular Pieces of Umbrella Cloth & Cost

Umbrella

6. An umbrella is made by stitching 10 triangular pieces of cloth of two different colours equally, each piece measuring 20 cm, 50 cm and 50 cm.

(a) Write the formula to find the area of triangle whose sides a, b and c are given. [1]
(b) How much cloth of each colour is required for the umbrella? [2]
(c) Find the total cloth required to make the umbrella. [1]
(d) Find the total cost of clothes at the rate of 50 paise per cm². [1]

Question 7

Dining Room Carpeting Expenses

Carpet

7. The dining room in Rohit’s house is 24 ft long and 15 ft wide. The floor is covered with carpet pieces of size 9 ft × 5 ft.

(a) Find the cost of carpeting the floor if the rate of cost of carpet is Rs 30 per sq.ft. [2]
(b) Calculate the cost of carpeting the floor at Rs 1250 per piece. [2]
(c) If the rate of cost of carpet is Rs 160 per ft², find the cost of carpeting the floor. [1]

Question 8

Square Garden Path Area & Paving Cost

Path

8. A square garden is 40 m long. If a path with uniform width 2 m surrounds the garden:

(a) Find the area of the path. [2]
(b) Calculate the cost of paving stones on the path at the rate of Rs 350 per m². [1]
(c) Find the cost of maintaining the garden at the rate of Rs 500 per m². [2]

Question 9

Classroom Walls & Wallpaper Count

Wallpaper

9. A classroom has length 8 m, breadth 6 m and height 3 m. It has two doors of dimension 2 m × 1.5 m each and two windows of dimension 1.5 m × 1 m.

(a) Write the formula for area of 4 walls excluding the doors and windows. [1]
(b) Calculate the area of 4 walls excluding the doors and windows. [2]
(c) If the four walls are to be covered by wall papers of size 1 m² each, how many such wall papers are required? [1]

Question 10

Room Walls & Ceiling Painting Expenses

Painting

10. A room has length 15 ft, breadth 10 ft and height 8 ft. It has two square windows of length 3 ft each and two doors of dimension 6 ft × 2 ft each.

(a) Calculate the area of 4 walls and ceiling of the room without windows and doors.
(b) If the 4 walls and the ceiling are to be painted at the rate of Rs 175 per sq.ft, find the total cost of painting.