7 th maths chapter 2 || 2.3

7th Standard Mathematics — Chapter 2: Measurements

Exercise 2.3 Solutions (Trapezium)

1. Find the missing values

Formula: Area of Trapezium = ½ × h × (a + b)

S.No.	Height 'h'	Parallel side 'a'	Parallel side 'b'	Area
(i)	10 m	12 m	20 m	160 sq. m
(ii)	24 cm	13 cm	28 cm	492 sq. cm
(iii)	19 m	18 m	16 m	323 sq. m
(iv)	16 cm	15 cm	30 cm	360 sq. cm

Workings:

• (i) Area: ½ × 10 × (12 + 20) = 5 × 32 = 160 sq. m
• (ii) Height (h): 492 = ½ × h × (13 + 28) ⇒ 492 = ½ × h × 41 ⇒ h = (492 × 2) ÷ 41 = 24 cm
• (iii) Parallel side 'a': 323 = ½ × 19 × (a + 16) ⇒ (323 × 2) ÷ 19 = a + 16 ⇒ 34 = a + 16 ⇒ a = 18 m
• (iv) Parallel side 'b': 360 = ½ × 16 × (15 + b) ⇒ 360 = 8 × (15 + b) ⇒ 15 + b = 45 ⇒ b = 30 cm

2. Area Calculation

Find the area of a trapezium whose parallel sides are 24 cm and 20 cm and the distance between them is 15 cm.

Solution:

• Parallel sides: a = 24 cm, b = 20 cm
• Distance between them (height, h) = 15 cm

Area = ½ × h × (a + b)

Area = ½ × 15 × (24 + 20) = ½ × 15 × 44 = 15 × 22 = 330 sq. cm

Answer: 330 sq. cm

3. Finding the Other Parallel Side

The area of a trapezium is 1586 sq. cm. The distance between its parallel sides is 26 cm. If one of the parallel sides is 84 cm then, find the other side.

Solution:

• Area = 1586 sq. cm
• Height (h) = 26 cm
• Side a = 84 cm

Area = ½ × h × (a + b)

1586 = ½ × 26 × (84 + b)

1586 = 13 × (84 + b)

84 + b = 1586 ÷ 13

84 + b = 122

b = 122 - 84 = 38 cm

Answer: The length of the other parallel side is 38 cm.

4. Finding the Height (Distance Between Parallel Sides)

The area of a trapezium is 1080 sq. cm. If the lengths of its parallel sides are 55.6 cm and 34.4 cm, find the distance between them.

Solution:

• Area = 1080 sq. cm
• a = 55.6 cm, b = 34.4 cm
• a + b = 55.6 + 34.4 = 90 cm

Area = ½ × h × (a + b)

1080 = ½ × h × 90

1080 = 45 × h

h = 1080 ÷ 45 = 24 cm

Answer: The distance between the parallel sides is 24 cm.

5. Finding Lengths of Both Parallel Sides

The area of a trapezium is 180 sq. cm and its height is 9 cm. If one of the parallel sides is longer than the other by 6 cm, find the length of the parallel sides.

Solution:

• Area = 180 sq. cm
• Height (h) = 9 cm
• Let smaller side 'b' = x
• Then longer side 'a' = x + 6

Area = ½ × h × (a + b)

180 = ½ × 9 × (x + 6 + x)

(180 × 2) ÷ 9 = 2x + 6

40 = 2x + 6

2x = 34 ⇒ x = 17 cm

• Smaller side (b) = 17 cm

• Longer side (a) = 17 + 6 = 23 cm

Answer: The lengths of the parallel sides are 23 cm and 17 cm.

6. Painting Cost Problem

The sunshade of a window is in the form of isosceles trapezium whose parallel sides are 81 cm and 64 cm and the distance between them is 6 cm. Find the cost of painting the surface at the rate of ₹ 2 per sq. cm.

Solution:

• a = 81 cm, b = 64 cm, h = 6 cm
• Area = ½ × h × (a + b) = ½ × 6 × (81 + 64) = 3 × 145 = 435 sq. cm
• Rate of painting = ₹ 2 per sq. cm

Total Cost Calculation:

Total Cost = 435 sq. cm × ₹ 2 = ₹ 870

Answer: The cost of painting the surface is ₹ 870.

7. Glass Covering Cost Problem

A window is in the form of trapezium whose parallel sides are 105 cm and 50 cm respectively and the distance between the parallel sides is 60 cm. Find the cost of the glass used to cover the window at the rate of ₹ 15 per 100 sq. cm.

Solution:

• a = 105 cm, b = 50 cm, h = 60 cm
• Area = ½ × h × (a + b) = ½ × 60 × (105 + 50) = 30 × 155 = 4650 sq. cm

Cost Calculation:

Rate = ₹ 15 per 100 sq. cm

Total Cost = (4650 ÷ 100) × ₹ 15 = 46.5 × ₹ 15 = ₹ 697.50

Answer: The cost of the glass used is ₹ 697.50.