Area of a Triangle | JHS Mathematics | The Let's Learn Project

Introduction

A triangle is a three-sided polygon with three angles. The area of a triangle is the amount of space enclosed within its three sides. Understanding how to calculate the area of triangles is important in many real-world applications.

Key Terms

Base (b): Any one side of the triangle, typically the bottom side when drawn
Height (h): The perpendicular distance from the base to the opposite vertex
Area: The amount of space inside the triangle, measured in square units

Note: The height must always be perpendicular to the base you choose.

The Area Formula

Basic Formula

The area of any triangle can be calculated using this formula:

\[ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \] \[ A = \frac{1}{2} \times b \times h \]

Why \(\frac{1}{2}\)?

A triangle is half of a parallelogram with the same base and height:

Finding the Height

Sometimes you need to identify the height in different triangle orientations:

The height can be inside, outside, or one of the sides of the triangle.

Calculations

Calculating Area

Follow these steps to calculate the area of a triangle:

Identify the base and corresponding height
Multiply the base by the height
Divide the result by 2

Example 1: Right Triangle

Base = 6 cm, Height = 4 cm

\[ A = \frac{1}{2} \times 6 \times 4 = \frac{24}{2} = 12 \text{ cm}^2 \]

Example 2: Acute Triangle

Base = 10 m, Height = 7 m

\[ A = \frac{1}{2} \times 10 \times 7 = \frac{70}{2} = 35 \text{ m}^2 \]

Different Triangle Types

The area formula works for all types of triangles:

Type	Example
Right Triangle	\(\frac{1}{2} \times 3 \times 4 = 6\)
Acute Triangle	\(\frac{1}{2} \times 5 \times 8 = 20\)
Obtuse Triangle	\(\frac{1}{2} \times 6 \times 4 = 12\)

Finding Missing Dimensions

You can rearrange the formula to find missing base or height:

\[ \text{If } A = \frac{1}{2}bh, \text{ then:} \] \[ b = \frac{2A}{h} \quad \text{and} \quad h = \frac{2A}{b} \]

Example: Area = 24 cm², Base = 8 cm, find height:

\[ h = \frac{2 \times 24}{8} = \frac{48}{8} = 6 \text{ cm} \]

Applications

Practical Examples

Calculating triangle areas is useful in many real-world situations:

Construction

Calculating roofing materials for triangular sections:

\[ \text{Roof section: } b = 12\text{m}, h = 4\text{m} \] \[ A = \frac{1}{2} \times 12 \times 4 = 24 \text{ m}^2 \]

Land Surveying

Measuring triangular plots of land:

\[ \text{Plot: } b = 50\text{m}, h = 30\text{m} \] \[ A = \frac{1}{2} \times 50 \times 30 = 750 \text{ m}^2 \]

Art and Design

Calculating fabric needed for triangular flags:

\[ \text{Flag: } b = 1.2\text{m}, h = 0.8\text{m} \] \[ A = \frac{1}{2} \times 1.2 \times 0.8 = 0.48 \text{ m}^2 \]

Practice Exercise

Question 1

Find the area of a triangle with base 8 cm and height 5 cm.

\[ A = \frac{1}{2} \times 8 \times 5 = 20 \text{ cm}^2 \]

Question 2

A triangle has an area of 36 m² and a height of 9 m. What is its base?

\[ b = \frac{2 \times 36}{9} = 8 \text{ m} \]

Question 3

Calculate the area of a right triangle with legs measuring 6 cm and 8 cm.

\[ A = \frac{1}{2} \times 6 \times 8 = 24 \text{ cm}^2 \]

Question 4

A triangular garden has a base of 12 m and an area of 42 m². Find its height.

\[ h = \frac{2 \times 42}{12} = 7 \text{ m} \]

Question 5

Two triangles have the same base of 10 cm. One has height 6 cm, the other 8 cm. What is the difference in their areas?

\[ A_1 = \frac{1}{2} \times 10 \times 6 = 30 \text{ cm}^2 \] \[ A_2 = \frac{1}{2} \times 10 \times 8 = 40 \text{ cm}^2 \] \[ \text{Difference} = 40 - 30 = 10 \text{ cm}^2 \]

Question 6

A triangular flag has an area of 0.6 m² and height of 1.2 m. Find its base length.

\[ b = \frac{2 \times 0.6}{1.2} = 1 \text{ m} \]

Ready for more challenges?

Full Practice Set

Math Challenge

Timed Test in 7 Levels

Test your skills with calculating triangle areas