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# help

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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 ]

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