A traffic signal board, indicating ‘SCHOOL AHEAD’, is an equilateral triangle with side ‘a’. Find the area of the signal board, using Heron’s formula. If its perimeter is 180 cm, what will be the area of the signal board?
Solution and Explanation
Side of traffic signal board = a
Perimeter of the signal board = 3a = 180 cm
∴ a = 60 cm
Semi perimeter of the signal board (s) = \(\frac{3a}{2}\)
By using Heron’s formula,
Area of triangle =\(\sqrt{\text{[s(s - a)(s - b)(s - c)]}}\)
Area of given triangle
= \(\sqrt{\text{[s(s - a)(s - b)(s - c)]}}\)
=\( \sqrt{\text{[s(s - a)(s - a)(s - a)]}}\)
= \(\text{(s - a)} \sqrt{\text{[s(s - a)]}}\)
since s = \(\frac{3a}{2}\)
\((\frac{3a}{2} - a)\sqrt{\frac{3a}{2}(\frac{3a}{2} - a)}\)
\(= (\frac{a}{2}) \sqrt{\frac{3a}{2}(\frac{a}{2})}\)
\(= \frac{a}{2} × \frac{a}{2} × \sqrt3\)
=\( (\frac{\sqrt3}{4})a^2\)\( .......(1)\)
Area of the signal board = \( (\frac{\sqrt3}{4})a^2\) sq. units
perimeter = 180 cm
side of triangle = \(\frac{180}{3}\) cm
a = 60 cm
Area of the signal board = \( (\frac{\sqrt3}{4})(60)^2\)
\(= \)\( (\frac{\sqrt3}{4})(3600)\)
\(= 900\sqrt3\)
Area of the signal board \(= 900\sqrt3\) cm2
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