Area of a triangle - Heron's formula — lesson. Mathematics State Board, Class 9.

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1. The formula used to calculate area of a triangle when three sides are known is called Heron's formula.

2. Heron's formula is derived by the Egypt mathematician Heron.

3. Heron's formula is also called as Hero's formula.

Area of a triangle \(=\)

\sqrt{s (s - a) (s - b) (s - c)}

, where, \(a\), \(b\) and \(c\) are the sides of the triangle and

s = \frac{a + b + c}{2}

, that is, \(s\) \(=\) semi-perimeter (or) half of the perimeter of the triangle.

Let us find the area of a triangle using Heron's formula.

Example:

Find the area of a triangle of sides \(14 \ cm\), \(10 \ cm\) and \(12 \ cm\).

Solution:

Let \(a =\) \(14 \ cm\), \(b =\) \(10 \ cm\) and \(c =\) \(12 \ cm\).

s = \frac{a + b + c}{2}

\(s\) \(=\)

\frac{(14 + 10 + 12)}{2}

\(s\) \(=\)

\frac{36}{2}

\(s\) \(=\) 18

Area of a triangle:

\(=\)

\sqrt{s (s - a) (s - b) (s - c)}

\(=\)

\sqrt{18 (18 - 14) (18 - 10) (18 - 12)}

\(=\)

\sqrt{18 (4) (8) (6)}

\(=\)

\sqrt{6 \times 3 \times 4 \times 4 \times 2 \times 6}

\(=\)

\sqrt{6^{2} \times 4^{2} \times 3 \times 2}

\(=\)

6 \times 4 \sqrt{3 \times 2}

\(=\)

24 \sqrt{6}

Area of a triangle \(=\)

24 \sqrt{6}

\(cm^2\).

Did you find an error?

2. Area of a triangle - Heron's formula