Mensuration: Area and Perimeter of Plane Figures

Overview

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

This topic carries significant weight in the mathematics section and appears consistently in TET examinations. Questions typically involve direct formula application, word problems related to real-life situations (fencing a field, tiling a floor, painting a wall), and composite figures combining two or more shapes. Mastery requires memorizing formulas and understanding when to apply each.

Students must be comfortable with basic arithmetic operations, unit conversions, and visualizing shapes. The pedagogy component may also ask how to teach mensuration concepts to primary students using concrete materials like tiles, paper cutouts, and grid sheets.

Key Concepts

• **Perimeter** is the total length of the boundary of a closed figure. It is measured in linear units (cm, m, km).

• **Area** is the amount of surface enclosed within a closed figure. It is measured in square units (cm², m², km²).

• **Unit consistency** is critical—all measurements must be in the same unit before calculation. Convert cm to m or vice versa as needed.

• **Composite figures** are shapes formed by combining simple figures. Find the area by adding areas of component shapes or subtracting the removed portion.

• **Relationship between perimeter and area**: Two figures can have the same perimeter but different areas, and vice versa. They are independent measures.

• **Square and rectangle** are the most frequently tested shapes, followed by triangle, parallelogram, and circle.

• For **circles**, understand the difference between radius (centre to boundary) and diameter (diameter = 2 × radius).

Formulas / Key Facts

### Rectangle

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

### Square

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

### Triangle

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

### Parallelogram

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

### Rhombus

• Perimeter = 4 × side = 4a
• Area = ½ × d₁ × d₂ (where d₁ and d₂ are diagonals)

### Trapezium

• Area = ½ × (sum of parallel sides) × height = ½ × (a + b) × h

### Circle

• Circumference (perimeter) = 2πr = πd (where π ≈ 22/7 or 3.14)
• Area = πr²

### Semicircle

• Perimeter = πr + 2r = r(π + 2)
• Area = πr²/2

### Useful Conversions

• 1 m = 100 cm
• 1 km = 1000 m
• 1 m² = 10,000 cm²
• 1 hectare = 10,000 m²

Worked Examples

**Example 1: Rectangle Problem**

A rectangular playground is 50 m long and 30 m wide. Find its perimeter and the cost of fencing at Rs 25 per metre.

*Solution:*

• Perimeter = 2(l + b) = 2(50 + 30) = 2 × 80 = 160 m
• Cost of fencing = 160 × 25 = Rs 4,000

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**Example 2: Area of Triangle**

Find the area of a triangle with base 12 cm and height 8 cm.

*Solution:*

• Area = ½ × base × height
• Area = ½ × 12 × 8 = 48 cm²

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**Example 3: Circle Problem**

A circular garden has radius 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 × 2 × 14 = 616 m²

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**Example 4: Composite Figure**

A rectangular room 8 m × 6 m has a square carpet of side 4 m in the centre. Find the area of the uncovered floor.

*Solution:*

• Area of room = 8 × 6 = 48 m²
• Area of carpet = 4 × 4 = 16 m²
• Uncovered area = 48 − 16 = 32 m²

Common Mistakes

• **Confusing perimeter and area formulas** → Remember: perimeter is the boundary (linear), area is the surface (square units). Perimeter of rectangle uses addition (l + b), area uses multiplication (l × b).

• **Forgetting to square the radius for circle area** → Students write πr instead of πr². Always check: circumference has r, area has r².

• **Mixing units without conversion** → If length is in metres and breadth in centimetres, convert both to the same unit first. A 2 m × 50 cm rectangle has area = 2 × 0.5 = 1 m², not 100.

• **Using diameter instead of radius in circle formulas** → The standard formulas use radius. If diameter is given, divide by 2 first.

• **Forgetting to add the straight edge in semicircle perimeter** → Semicircle perimeter is not just half the circumference; it includes the diameter: πr + 2r.

• **Applying Heron's formula incorrectly** → Students forget to calculate s (semi-perimeter) first or make errors in the subtraction (s−a), (s−b), (s−c).

Quick Reference

• Rectangle: P = 2(l+b), A = l×b
• Square: P = 4a, A = a²
• Triangle: P = a+b+c, A = ½×base×height
• Circle: C = 2πr, A = πr²
• Parallelogram: A = base × height
• Always convert to same units before calculating
• Perimeter → linear units (m, cm); Area → square units (m², cm²)