SOLUTION: What is the area of an equilateral triangle with apothem 5.8cm?

Question 192605: What is the area of an equilateral triangle with apothem 5.8cm?
Answer by RAY100(1637) (Show Source): You can put this solution on YOUR website!
please draw sketch of equilateral triangle
from center draw lines to each vertex.
now sketch apothem as altitude to center from mid point of each side
the equilateral triangle now has 3 equal triangles inside. Each of these is divided into 2 equal triangles by the apothem
the central angle of each of these triangles is 60 deg, the angle of apothem to side is 90 deg, and the other angle is 30.
this makes a 30-60-90 triangle, with small leg as apothem =5.8 cm.
the standard proportions of the 30-60-90 makes the longer leg = 5.8sqrt3.
this longer leg is 1/2 of a side, therefore side =2*5.8sqrt3=11.6sqrt3=20.09
the perimeter of base equilateral triangle is 3* 20.09 or= 60.275
remember for any polygon area = 1/2 * per* apothem=1/2*60.275*5.8=174.798
check using Heron's formula
sp=1/2 * 60.275 =30.1375
30.1375-20.09=10.05
A=sqrt( ( 30.1375)(10.05)^3)=174.84 ok
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