Area of a Triangle

Last updated on 26 January 2025

Table of contents

What is area of a triangle
Area of a Triangle Formula
Area of Triangle Using Heron's Formula
Area of a Triangle Given Two Sides and the Included Angle (SAS)
Area of a Right-Angled Triangle
Area of an Equilateral Triangle
Area of an Isosceles Triangle
Area of an Isosceles Right Triangle

The world of geometry is adorned with fundamental shapes, and the triangle stands proudly at the forefront. Beyond its simple appearance it has numerous properties, and one of the most essential aspects is its area. Understanding and calculating the area of a triangle is crucial in various fields, including mathematics, physics, engineering, architecture, and geography, where accurate measurements and spatial considerations are essential.

What is area of a triangle

The area of a triangle is the measure of the amount of space it occupies in a two-dimensional plane. For any given triangle, the area is expressed in square units. The concept of the area of a triangle is fundamental in geometry and is used to quantify the extent of a triangular region.

Area of a Triangle Formula

The area of a triangle can be calculated using various formulas. However, the basic formula that is used to find the area of a triangle is:

Area of Triangle = \frac{1}{2} \times base \times height

Example: Find the area of a triangle if it's base is 6 cm and height is 3 cm.

Given: $base = 6 cm$ , $height = 3 cm$

To find: Area of triangle

Solution: $\begin{array}{rcl} Area of Triangle & = & \frac{1}{2} \times base \times height \\ = & \frac{1}{2} \times 6 \times 3 \\ = & 3 \times 3 \\ = & 9 {cm}^{2} \end{array}$ Therefore, the area of the given triangle is $9 {cm}^{2}$ .

Area of Triangle Using Heron's Formula

For triangles with all three side lengths known, Heron's Formula provides an alternative method to calculate the area.

Consider the triangle above with side lengths

a

b

and

c

Semi Perimeter, s = \frac{a + b + c}{2}

Heron's formula:

Area = \sqrt{s (s - a) (s - b) (s - c)}

Example: Find the area of a triangle with side lengths 5 cm, 4 cm and 7 cm.

Given: $a = 5 cm$ , $b = 4 cm$ , $c = 7 cm$

To find: Area of triangle

Solution: First we will find the semi-perimeter of the triangle. $\begin{array}{lrcl} ∵ & s & = & \frac{a + b + c}{2} \\ ∴ & s & = & \frac{5 + 4 + 7}{2} \\ ∴ & s & = & \frac{16}{2} \\ ∴ & s & = & 8 cm \end{array}$ Now, by using Heron's formula, $\begin{array}{lrcl} ∵ & Area & = & \sqrt{s (s - a) (s - b) (s - c)} \\ ∴ & Area & = & \sqrt{8 (8 - 5) (8 - 4) (8 - 7)} \\ ∴ & Area & = & \sqrt{8 \times 3 \times 4 \times 1} \\ ∴ & Area & = & \sqrt{96} \\ ∴ & Area & = & \sqrt{16 \times 6} \\ ∴ & Area & = & \sqrt{4^{2} \times 6} \\ ∴ & Area & = & 4 \sqrt{6} {cm}^{2} \end{array}$

Therefore, the area of the given triangle is

4 \sqrt{6} {cm}^{2}

Area of a Triangle Given Two Sides and the Included Angle (SAS)

When the two sides of a triangle and an angle included between them are given as shown in the figure above, then the formula to calculate the area of a triangle is given by:

Area of triangle = \frac{1}{2} \times a \times b \times \sin θ

where

a

and

b

are the two sides of the triangle and

θ

is the angle included between them.

Example: Find the area of a triangle if two of its sides measure 4 cm and 5 cm, and angle included between those sides is 30°.

Given: $a = 4 cm$ , $b = 5 cm$ , $θ = 30 °$

To find: Area of triangle

Solution: $\begin{array}{lrcl} ∵ & Area & = & \frac{1}{2} \times a \times b \times \sin θ \\ ∴ & Area & = & \frac{1}{2} \times 4 \times 5 \times \sin 30 ° \\ ∴ & Area & = & 2 \times 5 \times \sin 30 ° \\ ∵ \sin 30 ° = \frac{1}{2} & ∴ & Area & = & 2 \times 5 \times \frac{1}{2} \\ ∴ & Area & = & 5 {cm}^{2} \end{array}$

Therefore, the area of the given triangle is

5 {cm}^{2}

Area of a Right-Angled Triangle

A right triangle has one angle that measures 90°, known as the right angle. The area can be calculated using the basic area formula with the lengths of the two sides that form the right angle.

Area of Right Triangle = \frac{1}{2} \times base \times height

Example: Find the area of a right-angled triangle shown in the figure below.

Given:

base = 7 cm

height = 6 cm

Solution:

\begin{array}{lrcl} ∵ & A & = & \frac{1}{2} \times base \times height \\ ∴ & A & = & \frac{1}{2} \times 7 \times 6 \\ ∴ & A & = & 7 \times 3 \\ ∴ & A & = & 21 {cm}^{2} \end{array}

Therefore, the area of the given right triangle is

21 {cm}^{2}

Area of an Equilateral Triangle

An equilateral triangle has all three sides of equal length. The area of an equilateral triangle can be calculated using the formula below:

Area = \frac{\sqrt{3}}{4} \times a^{2}

where $a$ is the length of one side of the equilateral triangle.

Example: Find the area of an equilateral triangle with side length =

8 cm

Given:

a = 8 cm

Solution:

\begin{array}{lrcl} ∵ & A & = & \frac{\sqrt{3}}{4} \times a^{2} \\ ∴ & A & = & \frac{\sqrt{3}}{4} \times 8^{2} \\ ∴ & A & = & \frac{\sqrt{3}}{4} \times 64 \\ ∴ & A & = & 16 \sqrt{3} {cm}^{2} \end{array}

Therefore, the area of the given equilateral triangle is

16 \sqrt{3} {cm}^{2}

Area of an Isosceles Triangle

An isosceles triangle has at least two sides of equal length. The angles opposite the equal sides are also equal.

The area of an isosceles triangle can be calculated using the formula below:

Area = \frac{b}{2} \sqrt{a^{2} - \frac{b^{2}}{4}}

where

b

is the length of the base (the unequal side) and

a

is the length of one of the two equal sides of an isosceles triangle.

Example: Find the area of an isosceles triangle given the length of its equal sides is

4 cm

and base is

6 cm

Given: $a = 4 cm$ , $b = 6 cm$

\begin{array}{lrcl} ∵ & A & = & \frac{b}{2} \sqrt{a^{2} - \frac{b^{2}}{4}} \\ ∴ & A & = & \frac{6}{2} \times \sqrt{4^{2} - \frac{6^{2}}{4}} \\ ∴ & A & = & 3 \times \sqrt{16 - \frac{36}{4}} \\ ∴ & A & = & 3 \times \sqrt{16 - 9} \\ ∴ & A & = & 3 \times \sqrt{7} \\ ∴ & A & = & 3 \sqrt{7} {cm}^{2} \end{array}

Therefore, the area of the given isosceles triangle is

3 \sqrt{7} {cm}^{2}

Area of an Isosceles Right Triangle

An isosceles right triangle has two sides of equal length and two equal angles measuring 45° and one right angle measuring 90°.

The area of an isosceles right triangle can be calculated using the formula below:

Area = \frac{1}{2} \times a^{2}

where

a

is the length of one of the two equal sides of an isosceles right triangle.

Example: Find the area of an isosceles right triangle if the length of it's two equal sides is 4 cm.

Given: $a = 4 cm$

To find: Area of the given isosceles right triangle

Solution: $\begin{array}{lrcl} ∵ & A & = & \frac{1}{2} \times a^{2} \\ ∴ & A & = & \frac{1}{2} \times 4^{2} \\ ∴ & A & = & \frac{1}{2} \times 16 \\ ∴ & A & = & 8 {cm}^{2} \end{array}$

Therefore, the area of the given isosceles right triangle is

8 {cm}^{2}

Area of a Triangle

What is area of a triangle

Area of a Triangle Formula

Area of Triangle Using Heron's Formula

Area of a Triangle Given Two Sides and the Included Angle (SAS)

Area of a Right-Angled Triangle

Area of an Equilateral Triangle

Area of an Isosceles Triangle

Area of an Isosceles Right Triangle

Acute Triangle

Perimeter of a Triangle