Fundamental Facts

Perimeter is the total length around a closed figure when it is expanded into a linear form. It is measured in units such as centimetres (cm), meters (m), inches (in), or feet (ft).

The perimeter of a regular polygon (a polygon with all sides and angles equal) is calculated as:

\[ \text{Perimeter} = \text{Number of sides} \times \text{Length of one side} \]

For a quadrilateral, the perimeter is the sum of the lengths of all four sides.

Units of perimeter correspond to the units of the sides measured.

Introduction

Perimeter: The distance covered along the boundary of a closed figure when you go around it once. It is the length of the boundary.

A regular polygon is a closed figure with all sides and angles equal.

Formulas for perimeter of common shapes:

• Rectangle: \( P = 2 \times (l + b) \), where \( l \) is length and \( b \) is breadth.
• Square: \( P = 4 \times s \), where \( s \) is the side length.
• Equilateral triangle: \( P = 3 \times s \), where \( s \) is the side length.

Area: The amount of surface enclosed by a closed figure.

Formulas for area of common shapes:

• Rectangle: \( A = l \times b \)
• Square: \( A = s \times s = s^2 \)

Perimeter

Perimeter is the total distance around the boundary of a closed figure.

Applications of perimeter include:

• Fencing a field.
• Building a compound wall around a house.
• Preparing a sports track.

Perimeter of a Rectangle

Sum of all sides:

\[ P = l + b + l + b = 2(l + b) \]

Perimeter of Regular Shapes

Regular polygons have all sides and angles equal.

Examples: Square, equilateral triangle, regular pentagon.

Formulas:

• Square: \( P = 4 \times s \)
• Equilateral triangle: \( P = 3 \times s \)
• Regular pentagon: \( P = 5 \times s \)

Example 1

Problem: Akshi wants to put lace all around a rectangular table cloth that is 3 m long and 2 m wide. Find the length of the lace required.

Solution:

Given:

• Length \( l = 3 \) m
• Breadth \( b = 2 \) m

Length of lace required = Perimeter of the table cloth

\[ P = 2(l + b) = 2(3 + 2) = 2 \times 5 = 10 \text{ m} \]

Therefore, 10 meters of lace is required.

Area

Area is the amount of surface enclosed by a closed figure.

To estimate area by counting squares on grid paper, use these conventions:

• Ignore portions less than half a square.
• Count portions more than half a square as one full square.
• If exactly half a square, count as \( \frac{1}{2} \) square unit.

Area of a Rectangle

Formula:

\[ A = l \times b \]

Area of a Square

Formula:

\[ A = s \times s = s^2 \]

Keywords

• Perimeter: Distance covered along the boundary of a closed figure.
• Area: Amount of surface enclosed by a closed figure.

Example 2

Problem: A floor is 5 m long and 4 m wide. A square carpet of side 3 m is laid on the floor. Find the area of the floor that is not carpeted.

Solution:

Floor dimensions:

• Length \( l = 5 \) m
• Width \( w = 4 \) m

Area of floor:

\[ A_{floor} = l \times w = 5 \times 4 = 20 \text{ m}^2 \]

Carpet side length \( s = 3 \) m

Area of carpet:

\[ A_{carpet} = s^2 = 3 \times 3 = 9 \text{ m}^2 \]

Area of floor not carpeted:

\[ A_{not\ carpeted} = A_{floor} - A_{carpet} = 20 - 9 = 11 \text{ m}^2 \]

Practice Set

Level 1 – Easy

• Find the perimeter of a square with side 6 cm.
• Calculate the area of a rectangle with length 8 m and breadth 3 m.
• Find the perimeter of an equilateral triangle with side 5 cm.

Level 2 – Moderate

• A rectangular garden is 15 m long and 10 m wide. Find the length of the fence required to surround the garden.
• A square room has side 12 m. Find the area of the room.
• Find the area of a rectangular field measuring 25 m by 18 m.

Level 3 – Challenging

• A rectangular plot is 40 m long and 30 m wide. A path 2 m wide is constructed all around inside the plot. Find the area of the path.
• A square park has a perimeter of 200 m. Find its area.
• A triangular field is equilateral with side 20 m. Find its perimeter and area.

Answer Key

Level 1 – Easy

• Perimeter of square = \(4 \times 6 = 24 \text{ cm}\)
• Area of rectangle = \(8 \times 3 = 24 \text{ m}^2\)
• Perimeter of equilateral triangle = \(3 \times 5 = 15 \text{ cm}\)

Level 2 – Moderate

• Fence length = Perimeter of garden = \(2(15 + 10) = 2 \times 25 = 50 \text{ m}\)
• Area of room = \(12 \times 12 = 144 \text{ m}^2\)
• Area of field = \(25 \times 18 = 450 \text{ m}^2\)

Level 3 – Challenging

• Area of path:
  • Outer area = \(40 \times 30 = 1200 \text{ m}^2\)
  • Inner dimensions = \(40 - 2 \times 2 = 36 \) m and \(30 - 2 \times 2 = 26 \) m
  • Inner area = \(36 \times 26 = 936 \text{ m}^2\)
  • Area of path = Outer area - Inner area = \(1200 - 936 = 264 \text{ m}^2\)
• Square park side = \(\frac{200}{4} = 50 \text{ m}\), area = \(50^2 = 2500 \text{ m}^2\)
• Equilateral triangle perimeter = \(3 \times 20 = 60 \text{ m}\)
• Area of equilateral triangle = \(\frac{\sqrt{3}}{4} s^2 = \frac{\sqrt{3}}{4} \times 20^2 = 100\sqrt{3} \approx 173.21 \text{ m}^2\)

Quick Reference

Shape	Perimeter	Area
Rectangle	\(2(l + b)\)	\(l \times b\)
Square	\(4s\)	\(s^2\)
Equilateral Triangle	\(3s\)	\(\frac{\sqrt{3}}{4} s^2\)
Regular Pentagon	\(5s\)	Not covered

Glossary

• Perimeter: The total length around the boundary of a closed figure.
• Area: The measure of the surface enclosed within a closed figure.
• Regular Polygon: A polygon with all sides and all angles equal.
• Equilateral Triangle: A triangle with all three sides equal.
• Square: A regular quadrilateral with four equal sides and four right angles.
• Rectangle: A quadrilateral with opposite sides equal and four right angles.