Question 1083252
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<pre>
1.  Make a sketch.

    Draw the apothem in the regular octagon.

2.  The central angle is {{{alpha}}} = {{{360/8}}} = 45 degs.

    The acute angle in the right-angled triangle is {{{beta}}} = {{{45/2}}} = 22.5 degrees.


    From the right-angled triangle, you have {{{6/2}}} = {{{R*sin(beta)}}},   or

    R = {{{3/sin(22.5^o)}}}.


    You can find {{{sin(22.5^o)}}} using your calculator or using the formula

    {{{sin(22.5^o)}}} = {{{sqrt(2-sqrt(2))/2}}}.
</pre>

See the lessons

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/Polygons/The-side-length-of-a-regular-polygon-via-the-radius-of-the-circumscribed-circle.lesson>The side length of a regular polygon via the radius of the circumscribed circle</A>

and

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/Trigonometry-basics/Miscellaneous-Trigonometry-problems.lesson>Miscellaneous Trigonometry problems</A>

in this site.