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Express the area A of an equilateral triangle in terms of its perimeter p.

 Jun 5, 2020
 #1
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Express the area A of an equilateral triangle in terms of its perimeter p.

 

A = sqrt [ p/2 (p/2-a)³ ]     smiley

 Jun 5, 2020
 #2
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If the perimeter is \(p\), then the side length is \(\frac{p}{3}\). Draw the altitude to one of the sides. Then, it divides this triangle into two \(30-60-90\) right triangles. Using the properties of such triangles, the altutude is \(\frac{p\sqrt3}{6}\). Thus, the area is \(\frac{1}{2} \cdot \text{base} \cdot \text{height} = \frac{1}{2} \cdot \frac{p}{3} \cdot \frac{p\sqrt3 }{6} = \boxed{\frac{p^2 \sqrt3}{36}}\)

 Jun 5, 2020

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