Area of an equilateral triangle — lesson. Mathematics CBSE, Class 9.

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Area of an equilateral triangle using Heron's formula:

\(AB\) \(=\) \(BC\) \(=\) \(CA\) \(=\) \(a\)

That is, \(a\) \(=\) \(b\) \(=\) \(c\)

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

\(s\) \(=\)

\frac{a + a + a}{2}

\(=\)

\frac{3 a}{2}

Area of a triangle \(=\)

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

\(=\)

\sqrt{\frac{3 a}{2} (\frac{3 a}{2} - a) (\frac{3 a}{2} - a) (\frac{3 a}{2} - a)}

\(=\)

\sqrt{\frac{3 a}{2} (\frac{3 a - 2 a}{2}) (\frac{3 a - 2 a}{2}) (\frac{3 a - 2 a}{2})}

\(=\)

\sqrt{\frac{3 a}{2} (\frac{a}{2}) (\frac{a}{2}) (\frac{a}{2})}

\(=\)

\sqrt{3 \times {(\frac{a}{2})}^{2} \times {(\frac{a}{2})}^{2}}

\(=\)

\frac{a}{2} \times \frac{a}{2} \sqrt{3}

\(=\)

\frac{\sqrt{3}}{4} a^{2}

Therefore, the area of an equilateral triangle \(=\)

\frac{\sqrt{3}}{4} a^{2}

sq. units.

Example:

Calculate the area of an equilateral triangle of side \(4 \ cm\).

\(a\) \(=\) \(b\) \(=\) \(c\) \(=\) \(4 \ cm\)

Area of an equilateral triangle \(=\)

\frac{\sqrt{3}}{4} a^{2}

\(=\)

\frac{\sqrt{3}}{4} \times (4^{2})

\(=\)

\frac{\sqrt{3}}{4} \times 16

\(=\) \(\sqrt{3} \times 4\)

\(=\) \(1.73 \times 4\) [Since \(\sqrt{3} = 1.73\)]

\(=\) \(6.92\)

Therefore, the area of an equilateral triangle \(=\) \(6.92 \ cm^2\).

Did you find an error?

3. Area of an equilateral triangle