SOLUTION: An exterior angle of a regular polygon measures 18 degrees. How many sides does the polygon have? If the length of the side of the polygon is 2 units, what is the area of the polyg
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Question 1017819: An exterior angle of a regular polygon measures 18 degrees. How many sides does the polygon have? If the length of the side of the polygon is 2 units, what is the area of the polygon? There are 20 sides just need help on the last part.
Found 2 solutions by Alan3354, Theo:
Answer by Alan3354(69443) (Show Source): You can put this solution on YOUR website!
An exterior angle of a regular polygon measures 18 degrees. How many sides does the polygon have? If the length of the side of the polygon is 2 units, what is the area of the polygon? There are 20 sides just need help on the last part.
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360/18 = 20 sides.
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where n = # of side and s = side length
=~ 126.275 sq units
Answer by Theo(13342) (Show Source): You can put this solution on YOUR website!
there are 20 sides to the polygon.
each side measures 2 units.
the polygon is divided into 20 isosceles triangles with the base of each isosceles triangle equal to 2.
the central angle each of the isosceles triangles is the same as the external angle of the polygon which is 18 degrees.
the other two angles of the isosceles triangle are therefore equal to (180 - 18) / 2 = 162 / 2 = 81 degrees each.
this is because the sum of the angles of any triangle is equal to 180.
18 + 2 * 81 = 18 + 162 = 180.
if you know the formula for the area of an isosceles triangle, you can simply work from there.
if not, then you need to solve for the altitude of the triangle and then find the area.
we'll assume we don't know it and we'll solve for the altitude.
since the base is 2 and the central angle of the triangle is 18, we can divide the isosceles triangle into 2 right triangles with base of 1 and angle of 9.
the altitude for this triangle would be tan(9) = 1/x, where x is the altitude.
solve for x to get x = 1/tan(9) = 6.313751515.
the areaa of the isosceles triangle is equal to 1/2 * b * h which is equal to 1/2 * 2 * 6.313751515 which is equal to 6.313751515.
since there are 20 of these isosceles triangles that make up the area of the polygon, then the area of the polygon is 20 * 6.313751515 = 126.2750503 square units.
i verified with an online calculator that this result is accurate.
a picture of your polygon is shown below:
a picture of one of the twenty isosceles triangles is shown below:
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