Hard Level Worksheet

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

In this hard level, we'll tackle advanced problems, diagonals, complex applications, and cost calculations.

Let's master the most challenging area and perimeter concepts!

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

Area = baseside × heightwidth

Perfect! Height must be perpendicular to the base.

2. Write the formula for the area of a triangle.

Area = one halftwo × baseside × heightlength

Correct! Area = (1/2) × base × height.

3. Find the perimeter of a square with side 9 m.

Perimeter = 4 ×

= m

Excellent! Perimeter = 36 m.

4. Find the area of a rectangle with length 25 m and breadth 15 m.

Area = ×

= m²

Perfect! Area = 375 m².

5. Write the unit of area and perimeter.

Area: square unitslinear units (cm², m²)

Perimeter: linear unitssquare units (cm, m)

Correct! Different types of measurement.

6. Define parallelogram.

A parallelogram is a quadrilateral with bothone pairs of opposite sides parallelperpendicular

Excellent! Both pairs of opposite sides are parallel.

7. The area of a triangle is 60 cm², and its base is 15 cm. Find height.

(1/2) × 15 × h =

7.5 × h = 60

h = 60 ÷

h = cm

Great! Height = 8 cm.

8. Find the area of a square whose perimeter is 48 cm.

Perimeter = 4 × side =

Side = cm

Area = × = cm²

Perfect! Area = 144 cm².

9. Write the relation between area and perimeter for squares.

If side = s, Perimeter = 4ss²

Area = s²4s

If P = perimeter, Area = (P ÷ 4) squaredtimes 2

Excellent! Area = (P/4)².

10. Find the perimeter of a parallelogram with sides 12 cm and 8 cm.

Perimeter = 2 × ( + )

= 2 ×

= cm

Perfect! Perimeter = 40 cm.

Drag each concept to its correct category:

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

1. The area of a rectangle is 96 cm². If its length is 12 cm, find breadth and perimeter.

Area = l × b =

12 × b = 96

b = cm

Perimeter = 2( + ) = 2 × = cm

Perfect! Breadth = 8 cm, Perimeter = 40 cm.

2. The base and height of a parallelogram are 10 cm and 6 cm, respectively. Find its area.

Area = base × height

= ×

= cm²

Excellent! Area = 60 cm².

3. Find the area and perimeter of a square whose diagonal is 14√2 cm.

For square with diagonal d, side = d ÷ √2

Side = 14√2 ÷ √

Side = cm

Area = × = cm²

Perimeter = 4 × = cm

Perfect! Area = 196 cm², Perimeter = 56 cm.

4. A triangle has base 10 cm and height 12 cm. Find its area.

Area = (1/2) × base × height

= (1/2) × ×

= (1/2) ×

= cm²

Great! Area = 60 cm².

5. Find the area of a parallelogram whose base is 18 cm and height 9 cm.

Area = ×

= cm²

Perfect! Area = 162 cm².

6. The perimeter of a rectangle is 60 cm, and its breadth is 10 cm. Find length and area.

2(l + 10) =

l + 10 =

l = cm

Area = × = cm²

Excellent! Length = 20 cm, Area = 200 cm².

7. Find the perimeter of a triangle with sides 12 cm, 15 cm, and 9 cm.

Perimeter = + +

= cm

Great! Perimeter = 36 cm.

8. A parallelogram and a rectangle are on the same base and between the same parallels. Compare their areas.

Both have samedifferent base

Both have samedifferent height

Therefore, areas are equalunequal

Perfect! They have equal areas.

9. The base of a triangle is 18 cm and height is 10 cm. Find its area.

Area = (1/2) × ×

= (1/2) ×

= cm²

Excellent! Area = 90 cm².

10. Find the side of a square whose area is equal to that of a rectangle with length 16 cm and breadth 9 cm.

Rectangle area = × = cm²

Square area = side × side =

Side = cm

Perfect! Side of square = 12 cm.

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

1. A rectangular field is 60 m long and 45 m wide. a) Find its area. b) Find the cost of fencing it at ₹25 per metre.

(a) Area:

Area = l × b = ×

= m²

(b) Cost of fencing:

Perimeter = 2( + )

= 2 ×

= m

Cost = 210 ×

= rupees

Perfect! Area = 2700 m², Cost = ₹5250.

2. A parallelogram has base 20 cm and height 15 cm. Find its area and perimeter if the other side = 18 cm.

Area: base × height

= ×

= cm²

Perimeter: 2(base + other side)

= 2( + )

= 2 ×

= cm

Excellent! Area = 300 cm², Perimeter = 76 cm.

3. The perimeter of a rectangle is 64 cm. If its length is twice its breadth, find its area.

Let breadth = b, length =

Perimeter = 2(2b + b) =

2 × 3b = 64

6b =

b = 64 ÷

b = 10.67 cm (approximately )

Length = 2 × 10.67 = cm (approximately)

Area: 21.34 ×

= approximately cm²

Note: Exact values: b = 32/3, l = 64/3, Area = 2048/9 cm²

Great work with the calculation!

4. The area of a triangle is 84 cm², and its base is 14 cm. Find its height.

Area = (1/2) × base × height =

(1/2) × 14 × h = 84

7 × h =

h = 84 ÷

h = cm

Perfect! Height = 12 cm.

5. Find the area and perimeter of a square plot whose diagonal is 20√2 m.

For square, diagonal = side × √

20√2 = side × √2

Side = m

Area: × = m²

Perimeter: 4 × = m

Excellent! Area = 400 m², Perimeter = 80 m.

Part B: Objective Questions - Test Your Knowledge!

Answer these multiple choice questions:

6. The area of a parallelogram with base 10 cm and height 6 cm is:

(a) 30 cm² (b) 40 cm² (c) 50 cm² (d) 60 cm²

Correct! Area = 10 × 6 = 60 cm².

7. If base = 8 cm and height = 9 cm, area of triangle =

(a) 36 cm² (b) 40 cm² (c) 32 cm² (d) 72 cm²

Perfect! Area = (1/2) × 8 × 9 = 36 cm².

8. The ratio of areas of two squares whose sides are 2 cm and 6 cm =

(a) 1:3 (b) 1:6 (c) 1:9 (d) 2:6

Excellent! Areas: 4 and 36, ratio = 4:36 = 1:9.

9. If the area of square = 100 cm², side =

(a) 5 cm (b) 10 cm (c) 15 cm (d) 20 cm

Correct! Side = √100 = 10 cm.

10. A parallelogram with same base and height as a rectangle has:

(a) Equal area (b) Half area (c) Double area (d) Greater perimeter

Perfect! Same base and height means equal area.

🏆 Outstanding Achievement! You've Mastered Advanced Area and Perimeter!

Here's what you've conquered at the hard level:

• Diagonal Relationships:

  Square:

  • Diagonal = side × √2
  • Side = diagonal ÷ √2
  • If diagonal = d, Area = d²/2
  • Example: d = 14√2 cm → side = 14 cm

  Rectangle:

  • Diagonal² = length² + breadth²
  • Use Pythagoras theorem
  • Diagonals are equal in rectangle

• Cost Calculations:

  Fencing (uses perimeter):

  • Calculate perimeter
  • Multiply by rate per meter
  • Example: P = 210 m, rate = ₹25/m → Cost = ₹5250

  Flooring/Carpeting (uses area):

  • Calculate area
  • Multiply by rate per square meter
  • Example: A = 100 m², rate = ₹50/m² → Cost = ₹5000

  Painting (uses area):

  • Calculate wall area
  • Subtract door/window areas
  • Multiply by rate per square meter

• Complex Ratio Problems:

  Method:

  • Express dimensions in terms of ratio
  • Use given perimeter/area condition
  • Solve for variable
  • Calculate required values

  Example:

  • Length:breadth = 3:2, P = 100m
  • Let l = 3x, b = 2x
  • 2(3x + 2x) = 100
  • 10x = 100 → x = 10
  • l = 30m, b = 20m, A = 600m²

• Parallelogram vs Rectangle:

  Same base and height:

  • Both have equal area
  • Area = base × height for both
  • Perimeter may differ

  Key difference:

  • Rectangle: all angles 90°
  • Parallelogram: opposite angles equal
  • But area formula is same

• Advanced Triangle Problems:

  Finding dimensions:

  • Given area and base → height = (2 × area) ÷ base
  • Given area and height → base = (2 × area) ÷ height

  Area relationships:

  • Triangle = (1/2) × rectangle (same base & height)
  • Triangle = (1/2) × parallelogram (same base & height)

• Square Root Applications:

  Finding side from area:

  • Area = side²
  • Side = √area
  • Example: A = 144 cm² → side = 12 cm

  Perfect squares:

  • 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225...
  • Memorize for quick calculations

• Ratio of Areas:

  Squares:

  • If sides in ratio a:b
  • Areas in ratio a²:b²
  • Example: sides 2:6 → areas 4:36 = 1:9

  Similar shapes:

  • Areas proportional to square of corresponding sides
  • Perimeters proportional to sides

• Comprehensive Problem-Solving:

  Multi-step problems:

  1. Read entire problem carefully
  2. Identify what's given and what to find
  3. Draw diagram if needed
  4. Break into smaller steps
  5. Use appropriate formulas
  6. Calculate systematically
  7. Verify answer makes sense
  8. Check units

• Units and Conversions:

  Length:

  • 1 m = 100 cm = 1000 mm
  • 1 km = 1000 m

  Area:

  • 1 m² = 10000 cm²
  • 1 km² = 1000000 m²
  • 1 hectare = 10000 m²

  Always convert to same units before calculating!

• Real-World Applications:

  Agriculture:

  • Field area for crop planning
  • Fencing perimeter calculation

  Construction:

  • Flooring, tiling, painting
  • Material estimation
  • Cost budgeting

  Interior Design:

  • Carpet/flooring area
  • Wallpaper requirements
  • Furniture placement

  Land Management:

  • Plot area measurement
  • Boundary marking
  • Property valuation

• Advanced Formulas Summary:

  Rectangle:

  • Area = l × b
  • Perimeter = 2(l + b)
  • Diagonal = √(l² + b²)

  Square:

  • Area = side²
  • Perimeter = 4 × side
  • Diagonal = side × √2
  • Area = (diagonal²)/2

  Triangle:

  • Area = (1/2) × base × height
  • Perimeter = a + b + c

  Parallelogram:

  • Area = base × height
  • Perimeter = 2(a + b)

• Common Advanced Mistakes:

  • Confusing side with diagonal in square
  • Using slant height instead of perpendicular height
  • Wrong formula for shape
  • Forgetting to convert units
  • Arithmetic errors in multi-step problems
  • Not simplifying ratios
  • Incorrect rounding in decimal answers

• Verification Techniques:

  • Check if perimeter > each individual side
  • Check if area units are square units
  • Verify calculations with reverse operations
  • See if answer is reasonable for given context
  • Cross-check with alternative methods

Mastering area and perimeter is essential for architecture, engineering, agriculture, and everyday life!

Remember: Practice various problem types to build confidence and speed!

Key to success: Clear understanding of formulas, systematic approach, and careful calculations!