# Perimeter and Area

## Overview
In this chapter, students learn to understand, calculate, and apply concepts of **perimeter** and **area** of closed shapes such as rectangles, squares, triangles, and more complex figures. They explore the use of standard formulas, apply them in real-life contexts, and use grid-based estimation techniques. They also compare shapes with same perimeter but different areas and vice versa.

## Key Topics Covered

### 1. Understanding Perimeter
- **Definition**: Perimeter is the total length of the boundary of a closed figure.
- **Perimeter of a Polygon**: Sum of all its sides.
- **Standard Formulas**:
- Rectangle: `Perimeter = 2 × (length + breadth)`
- Square: `Perimeter = 4 × side`
- Triangle: `Perimeter = side1 + side2 + side3`

**Examples**:
- Tablecloth: For a 3m by 2m table, lace needed = `2 × (3 + 2) = 10 m`
- Square park: For side 75 m, 3 rounds = `3 × 4 × 75 = 900 m`

### 2. Perimeter in Real-Life Context
- Comparison of running tracks of different sizes.
- Analysis of regular vs. irregular paths.
- Use of units such as meters or centimeters depending on object size.

**Activities Include**:
- Estimating boundary lengths using paper cutouts.
- Using straight (s) and diagonal (d) lines in units, such as `6s + 3d`.

### 3. Perimeter of Regular Polygons
- **Definition**: A regular polygon has all sides and angles equal.
- **Examples**:
- Equilateral triangle: `Perimeter = 3 × side`
- Regular hexagon: `Perimeter = 6 × side`

### 4. Understanding Area
- **Definition**: Area is the amount of space enclosed within a figure.
- **Units**: Square units (e.g., sq m, sq cm).
- **Standard Formulas**:
- Rectangle: `Area = length × breadth`
- Square: `Area = side × side`

**Examples**:
- A square carpet of 3m on a 5m × 4m floor → Uncarpeted area = `20 – 9 = 11 sq m`
- Land with square flower beds removed → Remaining area = Total area – Bed area

### 5. Real-Life Problems Involving Area
- Finding land area for farming or gardening.
- Calculating tiling cost.
- Estimating number of coconut trees in a grove using space required per tree.

### 6. Area of Irregular Figures and Grid Estimation
- Use of graph/squared paper to estimate area.
- Rules for estimation:
- Full square = 1 sq unit
- More than half square = 1 sq unit
- Less than half = 0
- Exactly half = ½ sq unit

### 7. Area of a Triangle
- **Key Insight**: A triangle formed by cutting a rectangle diagonally is exactly half the area of the rectangle.
- **General Formula (inferred)**:
- Area of triangle = ½ × base × height
- **Composite Figures**: Area found by splitting into known shapes (rectangles and triangles).

### 8. Area and Perimeter Comparison
- Shapes can have:
- Same perimeter but different area
- Same area but different perimeter
- Activity using unit squares (e.g., 9 squares) to explore different perimeters.

### 9. Area and Perimeter in House Plans
- Real-world application: Finding missing dimensions and total area in floor plans.
- Comparison of two different house layouts using area and perimeter.

### 10. Area Maze Puzzles
- Missing side or area is deduced using known values and properties.
- Focuses on analytical reasoning using area calculations.

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## New Terms

| Term | Definition |
|------------------------|----------------------------------------------------------------------------|
| Perimeter | Total length around a closed figure |
| Area | Total surface enclosed inside a closed figure |
| Regular polygon | Shape with all sides and angles equal |
| Diagonal | Line joining opposite corners of a shape |
| Unit square | A square measuring 1 unit by 1 unit |
| Estimation | An approximate calculation or judgment |
| Grid paper | Paper with evenly spaced horizontal and vertical lines |

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## Practice Questions

### Easy (3)

1. **Find the perimeter of a rectangle with length 6 cm and breadth 4 cm.**
**Answer**: `2 × (6 + 4) = 20 cm`

2. **What is the area of a square of side 5 cm?**
**Answer**: `5 × 5 = 25 sq cm`

3. **How many meters will Toshi run if she completes 4 rounds of a rectangle of size 30 m × 20 m?**
**Answer**: One round = `2 × (30 + 20) = 100 m` → `4 × 100 = 400 m`

### Medium (2)

4. **Find the missing side of a triangle with perimeter 45 cm and two sides 15 cm and 10 cm.**
**Answer**: `45 – (15 + 10) = 20 cm`

5. **The area of a rectangular garden is 240 sq m and length is 20 m. Find the breadth.**
**Answer**: `240 ÷ 20 = 12 m`

### Difficult (3)

6. **A square and a rectangle have the same perimeter of 40 cm. If the rectangle's length is 12 cm, find its breadth and area.**
**Solution**:
Perimeter of rectangle = `2 × (l + b)`
`40 = 2 × (12 + b)` → `20 = 12 + b` → `b = 8 cm`
Area = `12 × 8 = 96 sq cm`

7. **A square plot has area 81 sq m. Find its perimeter.**
**Solution**:
Side = √81 = 9 m → Perimeter = `4 × 9 = 36 m`

8. **How many 25 sq m coconut trees can be planted in a 100 m × 50 m rectangular grove?**
**Area = 5000 sq m → 5000 ÷ 25 = 200 trees**

### Very Difficult (2)

9. **A piece of string 36 cm long is shaped into a square. What is the length of one side? What is its area?**
**Perimeter = 36 cm → side = 36 ÷ 4 = 9 cm**
Area = `9 × 9 = 81 sq cm`

10. **A house plan has a kitchen of 15 ft × 12 ft, a bedroom 15 ft × 15 ft, and a hall of 20 ft × 15 ft. What is the total area of these rooms?**
**Area = 180 + 225 + 300 = 705 sq ft**

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