Heron's Formula Class 9 ||Maths|| Chapter 10 NCERT Notes

Heron's Formula ||Maths|| Chapter 10 NCERT Notes

1. Area of a Triangle (Recap)

For a triangle with base $b$ and height $h$ , the area is given by:
$Area of triangle = \frac{1}{2} \times base \times height$
However, when the height is not known but the lengths of all three sides are given, Heron’s formula can be used to find the area.

2. Heron’s Formula

Heron’s formula allows us to calculate the area of a triangle when all three sides are known.
For a triangle with sides $a$ , $b$ , and $c$ , the area is given by:
$Area of triangle = \sqrt{s (s - a) (s - b) (s - c)}$
Where:
$a$ , $b$ , and $c$ are the lengths of the sides of the triangle.
$s$ is the semi-perimeter of the triangle, which is half the perimeter of the triangle.
The semi-perimeter $s$ is calculated as:
$s = \frac{a + b + c}{2}$

3. Steps to Apply Heron’s Formula

To find the area of a triangle using Heron’s formula:
Step 1: Find the semi-perimeter $s$ of the triangle:
$s = \frac{a + b + c}{2}$
Step 2: Use Heron’s formula to find the area:
$Area = \sqrt{s (s - a) (s - b) (s - c)}$

4. Examples

Example 1: Area of a Triangle

Find the area of a triangle with sides 7 cm, 8 cm, and 9 cm.
Step 1: Find the semi-perimeter:
$s = \frac{7 + 8 + 9}{2} = \frac{24}{2} = 12 cm$
Step 2: Apply Heron’s formula:
$Area = \sqrt{12 (12 - 7) (12 - 8) (12 - 9)}$ $Area = \sqrt{12 \times 5 \times 4 \times 3}$ $Area = \sqrt{720} \approx 26.83 {cm}^{2}$

Example 2: Area of a Right-Angled Triangle

In a right-angled triangle with sides 5 cm, 12 cm, and 13 cm, find the area using Heron’s formula.
Step 1: Find the semi-perimeter:
$s = \frac{5 + 12 + 13}{2} = \frac{30}{2} = 15 cm$
Step 2: Apply Heron’s formula:
$Area = \sqrt{15 (15 - 5) (15 - 12) (15 - 13)}$ $Area = \sqrt{15 \times 10 \times 3 \times 2} = \sqrt{900} = 30 {cm}^{2}$
The area calculated using Heron’s formula (30 cm²) is the same as using the standard formula for a right-angled triangle, $\frac{1}{2} \times base \times height = \frac{1}{2} \times 12 \times 5 = 30 {cm}^{2}$ .

5. Application to Quadrilaterals

Heron’s formula can also be used to find the area of a quadrilateral by dividing it into two triangles. For example, a quadrilateral with diagonals dividing it into two triangles can have its area calculated by finding the area of each triangle separately using Heron’s formula.

Example:

If a quadrilateral is divided into two triangles, each with known side lengths, apply Heron’s formula to each triangle and sum the areas to get the total area of the quadrilateral.

Summary:

Heron’s formula is a powerful tool for finding the area of a triangle when the lengths of all three sides are known.
The semi-perimeter is a key component of Heron’s formula, calculated by adding the sides and dividing by 2.
Heron’s formula can also be extended to find the area of quadrilaterals by dividing them into triangles.

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Heron's Formula Class 9 ||Maths|| Chapter 10 NCERT Notes

1. Area of a Triangle (Recap)

For a triangle with base $b$ and height $h$ , the area is given by:
$Area of triangle = \frac{1}{2} \times base \times height$
However, when the height is not known but the lengths of all three sides are given, Heron’s formula can be used to find the area.

2. Heron’s Formula

3. Steps to Apply Heron’s Formula

To find the area of a triangle using Heron’s formula:
Step 1: Find the semi-perimeter $s$ of the triangle:
$s = \frac{a + b + c}{2}$
Step 2: Use Heron’s formula to find the area:
$Area = \sqrt{s (s - a) (s - b) (s - c)}$

4. Examples

Example 1: Area of a Triangle

Example 2: Area of a Right-Angled Triangle

5. Application to Quadrilaterals

Heron’s formula can also be used to find the area of a quadrilateral by dividing it into two triangles. For example, a quadrilateral with diagonals dividing it into two triangles can have its area calculated by finding the area of each triangle separately using Heron’s formula.

Example:

If a quadrilateral is divided into two triangles, each with known side lengths, apply Heron’s formula to each triangle and sum the areas to get the total area of the quadrilateral.

Summary:

Heron's Formula Class 9 ||Maths|| Chapter 10 NCERT Notes

1. Area of a Triangle (Recap)

For a triangle with base bb and height hh, the area is given by:Area of triangle=12×base×heightArea of triangle=21​×base×heightHowever, when the height is not known but the lengths of all three sides are given, Heron’s formula can be used to find the area.

2. Heron’s Formula

3. Steps to Apply Heron’s Formula

To find the area of a triangle using Heron’s formula:Step 1: Find the semi-perimeter ss of the triangle:s=a+b+c2s=2a+b+c​Step 2: Use Heron’s formula to find the area:Area=s(s−a)(s−b)(s−c)Area=s(s−a)(s−b)(s−c)​

4. Examples

Example 1: Area of a Triangle

Example 2: Area of a Right-Angled Triangle

5. Application to Quadrilaterals

Heron’s formula can also be used to find the area of a quadrilateral by dividing it into two triangles. For example, a quadrilateral with diagonals dividing it into two triangles can have its area calculated by finding the area of each triangle separately using Heron’s formula.

Example:

If a quadrilateral is divided into two triangles, each with known side lengths, apply Heron’s formula to each triangle and sum the areas to get the total area of the quadrilateral.

Summary:

For a triangle with base $b$ and height $h$ , the area is given by:
$Area of triangle = \frac{1}{2} \times base \times height$
However, when the height is not known but the lengths of all three sides are given, Heron’s formula can be used to find the area.

To find the area of a triangle using Heron’s formula:
Step 1: Find the semi-perimeter $s$ of the triangle:
$s = \frac{a + b + c}{2}$
Step 2: Use Heron’s formula to find the area:
$Area = \sqrt{s (s - a) (s - b) (s - c)}$