Deriving Area of Equilateral Triangle

Topic Content:

An equilateral triangle has all three sides, s, equal in length.

We know that, Area of triangle = \(\frac{1}{2} \scriptsize \: \times \: base \: \times \: height\)

Where:

• base = s
• height = h

We can find an expression for h using Pythagoras theorem.

⇒ \(\scriptsize s^2 = h^2 \: + \: \left(\normalsize \frac{s}{2}\right) \scriptsize^2\)

⇒ \(\scriptsize h^2 = s^2 \: – \: \left(\normalsize \frac{s}{2}\right) \scriptsize^2\)

⇒ \(\scriptsize h^2 = s^2 \: – \: \normalsize \frac{s^2}{4} \)

⇒ \(\scriptsize h^2 = \normalsize \frac{4s^2 \: \: s^2}{4} \)

⇒ \(\scriptsize h^2 = \normalsize \frac{3 s^2}{4} \)

⇒ \(\scriptsize h = \sqrt{\normalsize \frac{3 s^2}{4}}\)

⇒ \(\scriptsize h = \normalsize \frac{1}{2} \scriptsize \left(\sqrt{3} s\right)\)

Now, put the value of “h” in the area