Area of Triangle for Class 12

Introduction

Triangles are among the most fundamental shapes in geometry, possessing a unique simplicity yet a wealth of mathematical properties. One of the key measurements associated with triangles is their area. The area of a triangle refers to the region enclosed within its three sides, and it holds significant importance in various fields, ranging from architecture and engineering to art and design. In this article, we will delve into the concept of the area of a triangle, explore the formula used to calculate it and uncover some practical applications of this fundamental geometric property.

What is the Area of Triangle?

A triangle is a polygon with three sides and three vertices, forming a closed shape. Each side connects two vertices, and the combination of these sides creates the enclosed space, which is referred to as the area of the triangle. Area is expressed in square units, such as square centimetres or square meters, depending on the unit of measurement used for the sides of the triangle.

Formula of Area of Triangle

The formula is as follows:

In this formula, the base represents the length of any one side of the triangle, and the height denotes the perpendicular distance from the base to the opposite vertex. It is essential to note that the height must be perpendicular to the base to obtain an accurate measurement of the triangle's area.

Let's explore how to use the formula to find the area of various types of triangles:

Equilateral Triangle

An equilateral triangle has all sides of equal length. Suppose we have an equilateral triangle with a side length of 6 centimetres. To find the area, we use the formula:

Area of an Equilateral Triangle =

Right-Angled Triangle

A right-angled triangle has one angle measuring 90^o (a right angle).

Area of a Right Triangle =

Isosceles Triangle

An isosceles triangle is a type of triangle where two of its sides are of equal length, and the angles opposite the equal sides are also equal.

Area of an Isosceles Triangle =

Where, b = base of the isosceles triangle
a = measure of equal sides of the isosceles triangle

Area of Triangle with Three Sides (Heron’s Formula)

Heron's formula is a method to calculate the area of a triangle when the lengths of all three sides are known. It is named after the ancient Greek mathematician Hero (also known as Heron) who first described this formula. Heron's formula provides a straightforward way to find the area of a triangle without needing to know the height or base length explicitly. The formula is as follows:

In this formula:
"s" represents the semi-perimeter of the triangle, which is calculated as half of the sum of the three sides:
s = (a + b + c) / 2
"a," "b," and "c" are the lengths of the three sides of the triangle.

To use Heron's formula, you need to know the lengths of all three sides of the triangle. Once you have the values for "a," "b" and "c," you can calculate the semi-perimeter "s." Then, plug the values into the formula to find the area of the triangle.