Hard Level Worksheet

Part A: Subjective Questions - Very Short Answer (1 Mark Each)

This advanced level focuses on reverse calculations, equilateral triangles, and complex rhombus problems.

Master these concepts to solve real-world geometric challenges!

1. Write the formula for the area of a parallelogram.

Area = base × height2(b + h)

Perfect! Area of parallelogram = base × height.

2. Find the height of a triangle whose base is 18 cm and area is 108 cm2.

Answer: cm

Excellent! Using Area = 12 × b × h, we get 108 = 1/2 × 18 × h, so h = 12 cm.

3. A rhombus has diagonals 18 cm and 24 cm. Find its area.

Answer: cm2

Correct! Area = 1/2 × d₁ × d₂ = 12 × 18 × 24 = 216 cm2.

4. Write the formula to find the area of an equilateral triangle.

Area = sqrt34a2sqrt34×a2 (Note: Either form is correct, where a is the side)

Great! Area of equilateral triangle = (√3/4) × side².

5. The area of a parallelogram is 72 cm2 and its base is 12 cm. Find its height.

Answer: cm

Perfect! Using Area = base × height, we get 72 = 12 × h, so h = 6 cm.

Drag each formula to its correct shape:

Part A: Section B – Short Answer Questions (2 Marks Each)

1. Find the area and perimeter of a rhombus whose diagonals are 30 cm and 16 cm.

Area = 12 × d₁ × d₂ = 12 × × = cm2

To find perimeter, first find the side using Pythagoras

Side = √[(d₁/2)² + (d₂/2)²] = √[15² + 8²] = √ = cm

Perimeter = 4 × side = 4 × 17 = cm

Excellent! Area = 240 cm2 and Perimeter = 68 cm.

2. The base of a triangle is 3 times its height. If its area is 96 cm2, find the base and height.

Let height = h, then base =

Area = 12 × base × height = 12 × 3h × h = h²

Given: 1.5h² = 96, so h² =

Height h = cm

Base = 3h = cm

Perfect! Base = 24 cm and height = 8 cm.

3. Find the area of a triangle whose sides are 8 cm, 15 cm, and 17 cm using Heron's formula.

Semi-perimeter s = (8 + 15 + 17) ÷ 2 = ÷ 2 = cm

s − a = 20 − 8 =

s − b = 20 − 15 =

s − c = 20 − 17 =

Area = √[20 × 12 × 5 × 3] = √ = cm2

Excellent! The area is 60 cm2.

4. The diagonals of a rhombus are 10 cm and 24 cm. Find its area and the side length of the rhombus.

Area = 12 × d₁ × d₂ = 12 × × = cm2

Side = √[(d₁/2)² + (d₂/2)²] = √[5² + 12²] = √[25 + 144] = √ = cm

Great! Area = 120 cm2 and side length = 13 cm.

5. A parallelogram has base 25 cm and an adjacent side 20 cm with an included angle of 60°. Find its area.

Area = base × height = base × (side × sin θ)

Area = 25 × (20 × sin 60°) = 25 × 20 ×

Area ≈ cm2 (round to nearest whole number)

Perfect! The area is approximately 433 cm2.

Part A: Section C – Long Answer Questions (4 Marks Each)

1. A field is in the shape of a trapezium with parallel sides 25 m and 15 m and the distance between them 12 m.

(a) Find its area.

Area = 12 × (a + b) × h = 12 × ( + ) ×

Area = 12 × × 12 = m²

Excellent! The area is 240 m².

(b) If 1 square metre of the field costs ₹150 to fence, find the total cost.

Note: The question asks for fencing cost per m² which is unusual. Typically fencing is per metre perimeter. Assuming it means cost per m² of area:__

Total cost = Area × rate = 240 ×

Total cost = ₹

Perfect! The total cost is ₹36,000.

2. The sides of a triangle are 9 cm, 12 cm, and 15 cm.

(a) Find its area using Heron's formula.

s = (9 + 12 + 15) ÷ 2 = ÷ 2 = cm

s − a = 18 − 9 =

s − b = 18 − 12 =

s − c = 18 − 15 =

Area = √[18 × 9 × 6 × 3] = √ = cm2

Great! The area is 54 cm2.

(b) Find the height corresponding to the smallest side.

The smallest side = cm (this is the base)

Using Area = 12 × base × height

54 = 12 × 9 × h

Height = (54 × 2) ÷ 9 = cm

Perfect! The height is 12 cm.

3. A parallelogram has base 36 cm and area 648 cm2.

(a) Find its height.

Area = base × height

648 = 36 × h

Height = 648 ÷ 36 = cm

Excellent! The height is 18 cm.

(b) If another parallelogram has the same base but double the height, find the ratio of their areas.

First parallelogram: Area₁ = 648 cm2

Second parallelogram: height = 2 × 18 = cm

Area₂ = 36 × 36 = cm2

Ratio = Area₁ : Area₂ = 648 : 1296 = :

Perfect! The ratio is 1 : 2.

4. A rhombus has a perimeter of 120 m and one diagonal measuring 48 m.

(a) Find its other diagonal.

Perimeter = 120 m, so each side = 120 ÷ 4 = m

Given diagonal d₁ = 48 m, so d₁/2 = m

Using Pythagoras: side2 = (d₁/2)² + (d₂/2)²

302 = 242 + d222

900 = 576 + d222

d222 =

d₂/2 = m, so d₂ = m

Great! The other diagonal is 36 m.

(b) Find its area.

Area = 12 × d₁ × d₂ = 12 × 48 × 36 = m2

Excellent! The area is 864 m2.

(c) Find the height of the rhombus if one of its sides makes an angle of 30° with the base.

Height = side × sin θ = 30 × sin 300 = 30 × = m

Perfect! The height is 15 m.

Part B: Objective Questions - Test Your Knowledge!

Answer these multiple choice questions:

6. The area of a trapezium =

(a) (a + b)h (b) 12(a + b)h (c) (a − b)h (d) 12(a − b)h

Perfect! Area of trapezium = 1/2(a + b)h, where a and b are parallel sides.

7. The height of a parallelogram having base 20 cm and area 160 cm2 =

(a) 4 cm (b) 6 cm (c) 8 cm (d) 10 cm

Correct! Height = Area ÷ base = 160 ÷ 20 = 8 cm.

8. The diagonals of a rhombus are 12 cm and 16 cm. Its area =

(a) 48 cm2 (b) 72 cm2 (c) 96 cm2 (d) 120 cm2

Excellent! Area = 1/2 × 12 × 16 = 96 cm2.

9. The height of a triangle having base 10 cm and area 25 cm2 =

(a) 4 cm (b) 5 cm (c) 6 cm (d) 7 cm

Perfect! Using Area = 1/2 × b × h, we get 25 = 1/2 × 10 × h, so h = 5 cm.

10. 1 hectare =

(a) 100 m² (b) 1000 m² (c) 10000 m² (d) 100000 m²

Correct! 1 hectare = 100 m × 100 m = 10000 m².

🎉 Exceptional Work! You've Mastered Advanced Area Concepts!

Here's what you learned:

• Reverse Calculations: Finding unknown dimensions (base, height) from given area

• Equilateral Triangle Formula: Special formula (√3/4)a² for equilateral triangles

• Complex Rhombus Problems: Finding diagonals using Pythagoras theorem and perimeter

• Heron's Formula Applications: Solving triangles when all three sides are known

• Parallelogram with Angles: Using trigonometry (sin θ) to find area

• Composite Problems: Multi-step problems combining multiple concepts

• Area Relationships: Understanding ratios between different shapes with same dimensions

• Real-world Applications: Field measurements, fencing costs, and practical geometry

These advanced skills prepare you for higher mathematics and complex real-world problem solving!