Question 165933
Question:  Find the length, to the nearest tenth, of the apothem of a regular octagon whose sides are each 10 inches long.
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Answer:  The center of a regular polygon is equidistant from the vertices.  The apothem is the distance from the center to a side.  A central angle of a regular polygon has its vertex at the center, and its sides pass through consecutive vertices.
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Each central angle measure of a regular n-gon is {{{360/n}}} degrees.
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Draw the octagon.  Draw an isosceles triangle with its vertex at the center of the octagon.  The central angle is {{{360/8}}} or 45 degrees.  Draw a segment that bisects the central angle and the side of the polygon to form a right triangle.
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Use the tangent ratio to find the apothem
{{{tan22.5=5/a}}}  The tangent of an angle is {{{"tangent_angle"="opposite_leg"/"adjacent_leg"}}}.
*NOTE: you use 22.5 because you bisected the central angle
{{{a=(5/(tan22.5))}}}  Solve for a.
a=8.96295... inches  Round to the nearest tenth
a=9.0 inches
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You can find a scanned picture of my work for this problem.  Just go to the solutions page and click on "apothem"
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