Perimeter, Area and Volume

Perimeter, Area and Volume

This topic covers the essential concepts of perimeter, area, and volume. Understanding these concepts is vital for solving problems in geometry and real-life applications, such as construction and design. You will learn how to calculate the perimeter and area of various shapes, as well as the volume of three-dimensional objects.

Perimeter of Polygons

The perimeter is the total length of the edges or sides of a polygon. For regular polygons, there are specific formulas to calculate the perimeter. For irregular polygons, measure each side and add the lengths together.

Formulas for Perimeter

Here are the formulas for calculating the perimeter of common polygons:

Polygon	Perimeter Formula
Triangle (scalene)	s1 + s2 + s3
Triangle (isosceles)	2s1 + s2
Triangle (equilateral)	3s
Quadrilateral (general)	s1 + s2 + s3 + s4
Rectangle	2(l + b)
Square	4s

Example 1: Calculating Perimeter

Calculate the perimeter of the following shapes:

• Scalene triangle with sides 5 m, 7 m, and 9 m.
• Rectangle with length 4 m and breadth 3 m.

Solution:

Triangle: Perimeter = 5 + 7 + 9 = 21 m

Rectangle: Perimeter = 2(4 + 3) = 2(7) = 14 m

Circumference of a Circle

The circumference is the perimeter of a circle. The formula for calculating the circumference is:

C = 2πr or C = πd

where r is the radius and d is the diameter. The value of π is approximately 3.14 or 22/7.

Example 2: Calculating Circumference

Calculate the circumference of a circle with a radius of 3 cm.

Solution:

C = 2πr = 2 × 3.14 × 3 = 18.84 cm

Area of Polygons

The area measures the space covered by a shape in square units. Different shapes have different formulas for calculating area.

Formulas for Area

Here are the formulas for calculating the area of common shapes:

Shape	Area Formula
Triangle	A = 1/2 × base × height
Rectangle	A = length × breadth
Square	A = side × side
Circle	A = πr²

Example 3: Calculating Area

Calculate the area of a rectangle with a length of 5 m and a breadth of 4 m.

Solution:

A = length × breadth = 5 × 4 = 20 m²

Volume of Three-Dimensional Objects

The volume measures the space occupied by a three-dimensional object. The volume is expressed in cubic units.

Formulas for Volume

Here are the formulas for calculating the volume of common solids:

Solid	Volume Formula
Rectangular prism	V = length × breadth × height
Cube	V = side³
Cylinder	V = πr²h
Sphere	V = 4/3πr³

Example 4: Calculating Volume

Calculate the volume of a cylinder with a radius of 2 cm and a height of 5 cm.

Solution:

V = πr²h = π × (2)² × 5 = 20π cm³ ≈ 62.83 cm³

Summary

• The perimeter is the total length of the sides of polygons.
• The area measures the space covered by a shape.
• The volume measures the space occupied by a three-dimensional object.

Check your understanding

1. What is the perimeter of a rectangle with a length of 10 m and a breadth of 4 m?
2. Calculate the area of a triangle with a base of 6 cm and a height of 4 cm.
3. What is the volume of a cube with a side length of 3 cm?
4. Find the circumference of a circle with a diameter of 10 cm.