Mensuration: Area and Perimeter of Simple Plane Figures

Overview

Mensuration is the branch of mathematics dealing with the measurement of geometric figures—their lengths, areas, and volumes. For Bihar TET Paper I, the focus is strictly on **plane figures** (2D shapes), specifically calculating their perimeter (boundary length) and area (surface covered).

This topic appears consistently in Bihar TET mathematics sections, typically carrying 2–4 questions. Questions range from direct formula application to word problems involving fencing, flooring, painting walls, or finding dimensions when area/perimeter is given. Mastery requires memorizing formulas and understanding when to apply each one.

Students must be comfortable with squares, rectangles, triangles, circles, and simple composite figures. The ability to visualize shapes and convert units (cm to m, m² to cm²) is equally essential for scoring full marks.

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Key Concepts

• **Perimeter** is the total length of the boundary of a closed figure. Think of it as the length of wire needed to fence a plot. Unit: metre (m), centimetre (cm).

• **Area** is the amount of surface enclosed by a figure. Think of it as the number of unit squares that fit inside the shape. Unit: square metre (m²), square centimetre (cm²).

• **Perimeter is a linear measure (one-dimensional); area is a square measure (two-dimensional).** This distinction matters when converting units.

• For **composite figures** (L-shaped rooms, pathways), break the shape into simpler figures, calculate separately, then add or subtract as needed.

• **Circumference** is the perimeter of a circle. The ratio of circumference to diameter is always π (pi), approximately 22/7 or 3.14.

• When a **path or border** surrounds a rectangle, the path area = Area of outer rectangle − Area of inner rectangle.

• **Unit conversion rule**: 1 m = 100 cm, so 1 m² = 10,000 cm². Always ensure consistent units before calculating.

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Formulas / Key Facts

### Square (side = a)

• Perimeter = 4a
• Area = a²
• Diagonal = a√2

### Rectangle (length = l, breadth = b)

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

### Triangle

• Perimeter = sum of all three sides (a + b + c)
• Area (general) = ½ × base × height
• Area (Heron's formula): √[s(s−a)(s−b)(s−c)], where s = (a+b+c)/2
• Equilateral triangle (side a): Area = (√3/4) × a²

### Circle (radius = r, diameter = d = 2r)

• Circumference = 2πr = πd
• Area = πr²

### Semicircle (radius = r)

• Perimeter = πr + 2r (curved part + diameter)
• Area = πr²/2

### Parallelogram (base = b, height = h)

• Perimeter = 2(a + b), where a and b are adjacent sides
• Area = b × h

### Rhombus (diagonals d₁ and d₂)

• Perimeter = 4 × side
• Area = ½ × d₁ × d₂

### Trapezium (parallel sides a and b, height h)

• Area = ½ × (a + b) × h

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Worked Examples

### Example 1: Rectangle — Finding Area and Perimeter **Problem:** A rectangular garden is 25 m long and 15 m wide. Find its perimeter and area.

**Solution:**

• Perimeter = 2(l + b) = 2(25 + 15) = 2 × 40 = **80 m**
• Area = l × b = 25 × 15 = **375 m²**

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### Example 2: Circle — Circumference and Area **Problem:** The radius of a circular park is 14 m. Find its circumference and area. (Take π = 22/7)

**Solution:**

• Circumference = 2πr = 2 × (22/7) × 14 = 2 × 22 × 2 = **88 m**
• Area = πr² = (22/7) × 14 × 14 = (22/7) × 196 = 22 × 28 = **616 m²**

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### Example 3: Path Around a Rectangle **Problem:** A rectangular field is 50 m by 40 m. A path 5 m wide runs around it outside. Find the area of the path.

**Solution:**

• Outer length = 50 + 5 + 5 = 60 m
• Outer breadth = 40 + 5 + 5 = 50 m
• Outer area = 60 × 50 = 3000 m²
• Inner area (field) = 50 × 40 = 2000 m²
• **Area of path = 3000 − 2000 = 1000 m²**

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### Example 4: Triangle Using Heron's Formula **Problem:** Find the area of a triangle with sides 13 cm, 14 cm, and 15 cm.

**Solution:**

• s = (13 + 14 + 15)/2 = 42/2 = 21 cm
• Area = √[s(s−a)(s−b)(s−c)]
• Area = √[21 × 8 × 7 × 6] = √[21 × 8 × 42] = √7056 = **84 cm²**

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Common Mistakes

| Wrong Thinking | Correct Fix | |----------------|-------------| | Confusing perimeter and area formulas—writing 4a for area of square. | Perimeter uses addition/multiplication by count of sides; area uses multiplication of dimensions. Perimeter has linear units (m), area has square units (m²). | | Forgetting to halve when calculating triangle area—writing base × height instead of ½ × base × height. | Always include the ½ factor for triangles. Visualize: a triangle is half of a parallelogram. | | Using diameter instead of radius in circle formulas, or vice versa. | Read the problem carefully. If diameter is given, divide by 2 to get radius before applying πr² or 2πr. | | Ignoring unit conversion—adding metres and centimetres directly. | Convert all measurements to the same unit first. Remember: 1 m² = 10,000 cm², not 100 cm². | | In path problems, adding path width only once instead of on both sides. | A path around a rectangle adds width to **both** ends of length and breadth. Add 2 × path width to each dimension. |

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Quick Reference

• **Square:** P = 4a, A = a²
• **Rectangle:** P = 2(l+b), A = l×b
• **Triangle:** A = ½ × base × height; use Heron's when height is unknown
• **Circle:** C = 2πr, A = πr²; use π = 22/7 unless told otherwise
• **Path area = Outer area − Inner area**
• **Always check units before calculating; convert if needed**