How do you find the apothem of an octagon with only the side length?

I wish I could draw or insert an octagon on this page.

You draw an octagon, or do a google search for octagon and copy paste the image of the octagon on MS Word. Set it so one side is horizontal. There should be another horizontal side on the top of the octogon.

Draw an x by drawing line from the left end point of the top side to the right end point of the bottom side. Draw another line from the right end point of the top side to the left end point of the bottom side. The 2 lines intersect at the center of the octagon. If you continue drawing these lines you will have 8 triangles, each with a vertex angle equal to 1/8 of 360o. So, the vertex angle of each of the 8 isosceles triangles is 45o

Look at the triangle whose base is the bottom side of the octagon The vertex angle at the top of this isosceles triangle is 45o. The sum of the interior angles of a triangle equals 180 o. The two base angles are equal. Let x equal the degrees of one of the base angles of the isosceles triangle.

2x + 45o = 180o

2x = 135o

x = 67.5o

Draw a line straight down from the center point of the octagon to the base of the triangle (bottom of octagon). This vertical line is the apothem. The apothem (vertical line) cuts the isosceles triangle into 2 right triangles. Now we can use trigonometry. Looking at the right triangle on the left side of the apothem, you know the left, bottom angle equals 67.5o and the apothem is the side opposite that angle.

Tangent = Opposite ÷ Adjacent

Adjacent is ½ of the length of the side of the octagon.

Opposite is the apothem

Example:

Side length = 10cm

½ side length = 5 cm

Angle = 67.5o

Tangent 67.5o = Opposite ÷ 5cm

Opposite = Tangent 67.5o × 5cm

Opposite = 12.07 cm

Or

Since the apothem is perpendicular to the base of the isosceles triangle and bisects the base, it also bisects the vertex angle in to two angles of 22.5o. Therefore, in the right triangle with base the one half of the side s of the octagon and height the apothem a we have:

a = (s/2)cot 22.5o

In the above example,

a = 5 cm (cot 22.5o) ≈ 12 cm