Question 1032156: Determine the area of a regular 18-gon with apothem 3. Use decimal approximations for your answer if needed
Found 2 solutions by Theo, Alan3354:
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! the polygon has 18 sides.
the central angle is therefore equal to 360 / 18 = 20 degrees.
the polygon is divided into 18 isosceles triangles whose vertex is equal to 20 degrees and whose altitude (the apothem) is equal to 3.
each of these isosceles triangle is divided into two equal right triangles with their central angles equal to 10 degrees.
solve for the base of each of these right triangle by using the tangent formula.
you get tan(10) = x/3, where x is the base and 3 is the altitude.
solve for x to get x = 3 * tan(10).
since the base of the isosceles triangle is 2x, then you get 2x = 6 * tan910).
the area of each of the isosceles triangl;e is equal to 1/2 * base * height.
this becomes 1/2 * 6 * tan(10) * 3 which results in the area of one of the isosceles triangles is equal to 1.586942826 square units.
since there are 18 of these isosceles triangles in the polygon, then multiply this by 18 to get area of the polygon with 18 sides is equal to 28.56497087 square units.
you can round this to as many desimal digits as required.
if 2, then 28.56
if 3, then 28.565
if 4, then 28.5650
if 5, then 28.56497
etc.,
this is what your polygon looks like:
https://upload.wikimedia.org/wikipedia/commons/thumb/d/d8/Regular_polygon_18_annotated.svg/220px-Regular_polygon_18_annotated.svg.png
each of those little triangle with a vertex angle of 20 degrees is an isosceles triangle.
the apothem is the altitude of each of them.
Answer by Alan3354(69443)  (Show Source):
You can put this solution on YOUR website! Determine the area of a regular 18-gon with apothem 3. Use decimal approximations for your answer if needed
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You can work this out using 36 right triangles.
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Or the formula:
 n = # of sides, a = apothem
Area = 18*9*tan(10)
Area =~ 28.565 sq units
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