Heron's Formula

Class 9 Mathematics

Learn how to calculate the area of a triangle when you know the lengths of all three sides.

What is Heron's Formula?

Heron's formula (also known as Hero's formula) is used to find the area of a triangle when we know the lengths of all three sides. It is named after Hero of Alexandria, a Greek engineer and mathematician who first documented it.

The Formula

Area = √[s(s - a)(s - b)(s - c)]

Where:

• a, b, c are the lengths of the three sides of the triangle
• s is the semi-perimeter of the triangle: s = (a + b + c)/2

Why Use Heron's Formula?

Typically, we calculate the area of a triangle using the formula: Area = ½ × base × height. However, sometimes we don't know the height of a triangle, but we do know all three side lengths. In such cases, Heron's formula is extremely useful.

Step-by-Step Calculation

1

Calculate the semi-perimeter: s = (a + b + c)/2

2

Calculate the differences: (s - a), (s - b), and (s - c)

3

Multiply the semi-perimeter by the three differences: s(s - a)(s - b)(s - c)

4

Take the square root of the product: √[s(s - a)(s - b)(s - c)]

Example

Find the area of a triangle with sides of lengths 5 cm, 6 cm, and 7 cm.

Solution:

1

Calculate semi-perimeter: s = (5 + 6 + 7)/2 = 18/2 = 9 cm

2

Calculate differences: s - a = 9 - 5 = 4 cm, s - b = 9 - 6 = 3 cm, s - c = 9 - 7 = 2 cm

3

Multiply: s(s - a)(s - b)(s - c) = 9 × 4 × 3 × 2 = 216

4

Take square root: √216 = 6√6 ≈ 14.7 cm²

So, the area of the triangle is approximately 14.7 cm².

Interactive Calculator

Enter the lengths of the three sides of your triangle to calculate its area:

Side a:

Side b:

Side c:

Result:

Practice Problems

1. Find the area of a triangle with sides 6 cm, 8 cm, and 10 cm.

2. Calculate the area of an equilateral triangle with side length 12 cm.

3. A triangle has sides 7 cm, 24 cm, and 25 cm. Is it a right triangle? Calculate its area.

4. Find the area of a triangle with sides 9 cm, 12 cm, and 15 cm.

Solutions:

1. s = (6+8+10)/2 = 12, Area = √[12(12-6)(12-8)(12-10)] = √[12×6×4×2] = √576 = 24 cm²

2. s = (12+12+12)/2 = 18, Area = √[18(18-12)(18-12)(18-12)] = √[18×6×6×6] = √3888 = 36√3 ≈ 62.35 cm²

3. Yes (7² + 24² = 25²), s = (7+24+25)/2 = 28, Area = √[28(28-7)(28-24)(28-25)] = √[28×21×4×3] = √7056 = 84 cm²

4. s = (9+12+15)/2 = 18, Area = √[18(18-9)(18-12)(18-15)] = √[18×9×6×3] = √2916 = 54 cm²

Important Notes

• Heron's formula works for all types of triangles: scalene, isosceles, and equilateral.
• The triangle inequality theorem must be satisfied: a + b > c, a + c > b, and b + c > a.
• For right triangles, Heron's formula will give the same result as ½ × base × height.
• Heron's formula is especially useful for scalene triangles where the height is not known.

Real-World Applications

Heron's formula has practical applications in:

• Construction and architecture
• Surveying and cartography
• Computer graphics and game development
• Fabric design and manufacturing

Heron's Formula - Class 9 Mathematics

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