Area & Perimeter of 2D Shapes

🎯 Learning Objective

Find the area and perimeter of squares, rectangles, triangles, parallelograms, rhombi, trapeziums and circles.

💡 Concept

• Square: Area = side², Perimeter = 4 × side, Diagonal = side × √2
• Rectangle: Area = length × breadth, Perimeter = 2(l + b), Diagonal = √(l² + b²)
• Triangle: Area = ½ × base × height; Equilateral = (√3/4) × side²
• Heron's formula: Area = √(s(s−a)(s−b)(s−c)), where s = (a+b+c)/2
• Parallelogram: Area = base × height; Rhombus: Area = ½ × d₁ × d₂
• Trapezium: Area = ½ × (sum of parallel sides) × height
• Circle: Area = πr², Circumference = 2πr (take π = 22/7)

🧮 Key Formulas

Square: A = a², P = 4a, diagonal = a√2

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Rectangle: A = l×b, P = 2(l+b)

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Equilateral triangle: A = (√3/4)a²

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Rhombus: A = ½ d₁d₂ ; Trapezium: A = ½(a+b)h

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Circle: A = πr², C = 2πr

✏️ Easy Example

Q. Find the area and perimeter of a rectangle 12 m long and 5 m wide.

1. Area = length × breadth = 12 × 5
2. Perimeter = 2(l + b) = 2(12 + 5) = 2 × 17

Answer: Area = 60 m², Perimeter = 34 m

🇮🇳 Real-Life Example

Buying floor tiles for a room, or turf for a railway station platform, always starts with the same step — calculating the area in square metres.

📝 Exam-Level Example

Q. Find the area of a triangle with sides 13 cm, 14 cm and 15 cm.

1. s = (13 + 14 + 15)/2 = 42/2 = 21
2. Area = √(21 × (21−13) × (21−14) × (21−15))
3. = √(21 × 8 × 7 × 6) = √7056

Answer: 84 cm²

📝 Exam-Level Example

Q. A circular park has radius 7 m. Find its area and circumference (π = 22/7).

1. Area = πr² = (22/7) × 7 × 7 = 22 × 7
2. Circumference = 2πr = 2 × (22/7) × 7 = 2 × 22

Answer: Area = 154 m², Circumference = 44 m

📝 Exam-Level Example

Q. The diagonals of a rhombus are 16 cm and 12 cm. Find its area and side length.

1. Area = ½ × d₁ × d₂ = ½ × 16 × 12 = 96
2. Half-diagonals 8 and 6 form a right triangle with the side
3. Side = √(8² + 6²) = √(64 + 36) = √100 = 10

Answer: Area = 96 cm², Side = 10 cm

📝 Exam-Level Example

Q. Find the area of a trapezium with parallel sides 20 cm and 12 cm and height 5 cm.

1. Area = ½ × (sum of parallel sides) × height
2. = ½ × (20 + 12) × 5 = ½ × 32 × 5

Answer: 80 cm²

🪄 Memory Trick

For the same perimeter, a circle has the largest area, and among quadrilaterals a square beats every rectangle. Squares and circles are the 'area champions'.

⚠️ Common Mistakes

• ❌ Using the slant side instead of the perpendicular height in triangle/parallelogram/trapezium area
• ❌ Forgetting to halve d₁ × d₂ in the rhombus area formula
• ❌ Mixing up area (square units) and perimeter (linear units)

🏆 Exam Tips

• ✅ Memorise the 13-14-15 → 84 and 3-4-5 triangles; they appear directly in exams
• ✅ Keep π = 22/7 and choose radii that are multiples of 7 for clean answers

📌 Summary

• Square A = a², Rectangle A = l×b, Triangle A = ½bh
• Rhombus A = ½d₁d₂, Trapezium A = ½(a+b)h
• Equilateral A = (√3/4)a²; Heron's uses s = perimeter/2
• Circle A = πr², C = 2πr