What is the area of an equilateral triangle with a side of ^@R \space cm^@?


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Answer:

^@\dfrac { R^2 \sqrt{ 3 } } { 4 } \space cm^2^@

Step by Step Explanation:
  1. As per Heron's formula, the area of a triangle with sides ^@a, b^@ and ^@c^@ and perimeter ^@2S = \sqrt{ S(S - a)(S - b)(S - c) } ^@
  2. Here, ^@a = b = c = R^@ and ^@S = \dfrac {3}{2} \times a = \dfrac { 3R } { 2 }.^@
  3. ^@ \begin{align} \text { Therefore, Area } & = \sqrt { \dfrac { 3R } { 2 } \times (\dfrac { 3R } { 2 } - R) \times (\dfrac { 3R } { 2 } - R) \times (\dfrac { 3R } { 2 } - R) }\\ & = \sqrt { \dfrac { 3R } { 2 } \times \dfrac { R } { 2 } \times \dfrac { R } { 2 } \times \dfrac { R } { 2 } } \\ & = \dfrac { R^2 \sqrt{ 3 } } { 4 } cm^2 \end{align}^@

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