Questions on Geometry: Length, distance, coordinates, metric length answered by real tutors!

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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 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 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
The tangent of an angle is .
*NOTE: you use 22.5 because you bisected the central angle
Solve for a.
a=8.96295... inches Round to the nearest tenth
a=9.0 inches
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Check out my website by clicking on my profile.
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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contact justin.sheppard.tech@hotmail.com with any questions

You can put this solution on YOUR website!
Edwin's solution:
Warning: MRperkins's solution is correct up to the last step.
He apparently mis-pressed something on his calculator and got
the wrong answer. Here is my complete solution with drawings:
I need help with a question could someone please help??

Find the length, to the nearest tenth, of the apothem of a regular octogon whose sides are each 10 inches long?

  
Draw the octagon, all sides of which are 10 inches.
I'll just indicate that the bottom side is 10:
 

Now temporarily, connect the vertices
to the center:

I did that just to show that each
of those 8 angles at the center are
° = °, so that if we
erase all but the bottom two, like this:



Now we know that the angle in the
above is °


Now draw in an apothem, the line from
the center to the midpoint of the bottom
 side, and label it .


Since the sides
of the octagon are 10 each, the two parts of
the bottom side are 5 each. Also the 45° angle
is bisected into two angles which are 22.5° each




So lets take away everything but
just this little right triangle:



Then we just do a little trig on that triangle:

The side opposite the 22.5° angle is 5 and the
side adjacent to it is a, so



Multiply both sides by a:



Divide both sides by tan(22.5°):





or, to the nearest tenth,

 inches.

Edwin