 # Area of Triangle

In a two-dimensional plane, the area of a triangle is defined as the total space filled by its three sides. The area of a triangle is equal to half the product of its base and height, therefore A = 1/2 b h is the fundamental formula. This formula works for any triangle, whether it’s a scalene triangle, an isosceles triangle, or an equilateral triangle. It’s important to remember that a triangle’s base and height are perpendicular to each other.

We will cover the area of triangle formulae for several sorts of triangles, as well as some instances, in this session.

## What is the Area of a Triangle?

The area of a triangle is the area contained by the triangle’s sides. The area of a triangle changes based on the length of the sides and the internal angles of the triangle. A triangle’s area is measured in square units such as m2, cm2, in2, and so on.

## Area of a Triangle Formula

Several formulae may be used to compute the area of a triangle. When we know the lengths of all three sides of a triangle, we can apply Heron’s formula to compute its area. When we know two sides and the angle produced between them, we may apply trigonometric functions to calculate the area of a triangle. However, the following is the basic formula for calculating the area of a triangle:

Area of triangle = 1/2 × base × height

Observe the following figure to see the base and height of a triangle.

Let us find the area of a triangle using this formula.

Example: What is the area of a triangle with base ‘b’ = 2 cm and height ‘h’ = 4 cm?

Solution: Using the formula: Area of a Triangle, A = 1/2 × b × h = 1/2 × 4 × 2 = 4 cm2

Triangles can be classified based on their angles as acute, obtuse, or right triangles. They can be scalene, isosceles, or equilateral triangles when classified based on their sides. Let us learn about the other ways that are used to find the area of triangles with different scenarios and parameters.

## Area of Triangle Using Heron’s Formula

When the lengths of the triangle’s three sides are known, Heron’s formula is used to calculate the area. To utilise this calculation, we need to know the triangle’s perimeter, which is computed by summing the lengths of all three sides and is the distance travelled around the triangle. There are two crucial phases in Heron’s formula.

Step 1: Add all three sides together and divide by two to get the semi perimeter (half perimeter) of the provided triangle.

Step 2: In the primary formula, ‘Heron’s Formula,’ plug in the value of the triangle’s semi-perimeter.

## Area of Triangle With 2 Sides and Included Angle (SAS)

In the case of a triangle with two sides and an angle, we use a formula that has three variations depending on the given dimensions. Here is an example.

When sides ‘b’ and ‘c’ and included angle A is known, the area of the triangle is:

Area (∆ABC) = 1/2 × bc × sin(A)

When sides ‘a’ and ‘b’ and included angle C is known, the area of the triangle is:

Area (∆ABC) = 1/2 × ab × sin(C)

When sides ‘a’ and ‘c’ and included angle B is known, the area of the triangle is:

Area (∆ABC) = 1/2 × ac × sin(B)

Example: In ∆ABC, angle A = 30°, side ‘b’ = 4 units, side ‘c’ = 6 units.

Area (∆ABC) = 1/2 × bc × sin A

= 1/2 × 4 × 6 × sin 30º

= 12 × 1/2 (since sin 30º = 1/2)

Area = 6 square units.

## How to Find the Area of a Triangle?

Triangle areas can be calculated using a variety of formulas based on the type and dimensions of the triangle.

### Area of Triangle Formulas

Below are the formulas for all types of triangles, such as the equilateral triangle, the right-angled triangle, and the isosceles triangle.

### Area of a Right-Angled Triangle

A right-angled triangle, often known as a right triangle, has one angle of 90 degrees and two acute angles that add up to 90 degrees. As a result, the triangle’s height equals the length of the perpendicular side.

A = 1/2 Base Height = Area of a Right Triangle

### Area of an Equilateral Triangle

A triangle with all sides equal is known as an equilateral triangle. The base is divided into two equal halves by a perpendicular traced from the triangle’s vertex to the base. We need to know the lengths of the sides of the equilateral triangle to compute its area.

Area of an Equilateral Triangle = A = (√3)/4 × side2

### Area of an Isosceles Triangle

An isosceles triangle has two of its sides equal and the angles opposite the equal sides are also equal.

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