Question 203384: what is the area of an equilateral triangle whose apothem is 8 yards?
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! to get the apothemn, you inscribe a circle inside the equilateral triangle.
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the radius of that circle will intersect each side exactly in the middle and will be perpendicular to each side at that point.
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if you draw a line from the center of that circle to each of the corners of the triange, then the center of that circle divides the equilateral triangle into 3 separate triangles. each one of those separate triangles has a side of the equilateral triangle as it's base and the lines drawn from the center of the circle to each corner of the triangle as it's other two sides.
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each one of those triangles has a 120 degree angle at the center of the circle and two 30 degree angles at each end.
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the apothem in each in each of those 3 separate triangles is the perpendicular bisector each of them.
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what you have is 6 separate right triangles formed from those 3 separate triangles that have one angle of 60 degrees, one angle of 90 degrees, and one angle of 30 degrees each.
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since the apothem is the height of each of those 6 right triangles, then the base of each of those 6 right triangles is equal to the apothem of each of those right triangles * cot(30) where 30 represents 30 degrees.
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this is derivede as follows:
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cot 30) = adjacent side / opposite side = base / apothem.
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cot(30) = 1/tan(30) = 1/.577350269 = 1.732050808
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cot(30) = base / apothem = base/8 where 8 is the length of the apothem.
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multiply both sides of this equation by 8 to get:
base of each right triangle = cot(30) * 8 = 1.732050808 * 8 = 13.85640646
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since base of each right triangle = 1/2 the length of each side of the equilateral triangle, then each side of the equilateral triangle equals 2 * 13.85640646 = 27.71281292
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the area of the equilateral triangle should be able to be derived in a couple of ways.
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one way is to get the area of each of the right triangles and multiply by 6.
area of each right triangle is 1/2 * 8 * 13.85640646 = 55.42562584 * 6 = 332.5537551
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another way is to use the formula for the area of an equilateral triangle which is  which becomes  .
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we get the same answer both ways which is a good sign.
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the apothem is 8 (given).
each side of the equiateral triangle is 27.71281292 yards long.
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your answer is:
area of the equilateral triangle is 332.5537551 square yards.
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to draw your own picture, do the following.
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equilateral triangle = ABC
D = center of inscribed circle.
DE is perpendicular bisector of BC
DF is perpendicular bisector of AC
DG is perpendicular bisector of AB
Draw DB, DC, DA
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your 3 separate triangles are:
BDC, BDA, ADC
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your 6 separate right triangles are:
BDE, CDE (from BDC)
BDG, ADG (from BDA)
CDF, ADF (from ADC)
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your apothems are:
DE, DF, DG
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if you need a picture, let me know and i'll work one up and send it to you.
try drawing your own first.
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