Heron's Formula Class 9th Notes - Free NCERT Class 9 Maths Chapter 12 Notes - Download PDF

NCERT notes Class 9 Maths Chapter 12 Heron's Formula

Triangle: A geometrical closed figure that has three sides, three angles and three corners. The three corners are also called the vertices of the triangle.

Types of Triangles

Triangles can be divided into two categories:
1. Based on sides
2. Based on angels

Based on Sides

1. Equilateral Triangle
2. Isosceles Triangle
3. Scalene Triangle

Equilateral Triangle:

A triangle in which all sides are equal is called an equilateral triangle.

Area of equilateral triangle = $\frac{\sqrt3}{4}a^2$

Perimeter of equilateral triangle = $3a$

Where,

a is the side of the triangle.

Example: Find the area and perimeter of the equilateral triangle, in which the length of the side is 10cm.

Area of equilateral triangle = $\frac{\sqrt3}{4}10^2$

= $25\sqrt3$$cm^2$

Perimeter of equilateral triangle = $3 × 10$

= $30$

Isosceles Triangle:

A triangle in which any two sides are equal is called an isosceles triangle.

Area of isosclales triangle = $\frac{1}{2}bh$

Perimeter of an isosceles triangle = $2a + b$

Example: Find the area and perimeter of the triangle whose base is 12 cm long and each equal side is 16 cm long.

Area of isosclales triangle = $\frac{1}{2}12 × 16$

Area of isosclales triangle = $96$

Perimeter of an isosceles triangle = $2 × 16 + 12$

Perimeter of an isosceles triangle = $44$

Scalene Triangle

A triangle where the length of each side is different is known as a scalene triangle.

Area of scalene triangle = $\frac{1}{2}bh$

Perimeter of an scalene triangle = $a + b + c$

Area of a Triangle by Heron's Formula

Heron is a mathematician and an engineer in Alexandria in Egypt, he defined a formula to determine the area of a triangle when only all three sides are known. The Heron's formula for determining the area of a triangle is as follows:

Semiperimeter (S) = $\frac{(a + b + c)}{2}$

Where a, b, and c are the sides of the triangle.

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

Example: Find the area of a triangle whose sides are 4 cm, 5 cm and 7cm respectively.

Semiperimeter (S) = $\frac{(a + b + c)}{2}$

Semiperimeter (S) = $\frac{(4 + 5 + 7)}{2}$

Semiperimeter (S) = $8$

Area of triangle = $\sqrt {8(8 - 4)(8 - 5)(8 - 7)}$

Area of triangle = $\sqrt {96}$$cm^2$

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