Question 957218
<pre>
{{{drawing(400,400,-1.2,1.2,-1.2,1.2,
line(0.92387953,0.38268343,0.38268343,0.92387953),
line(0.38268343,0.92387953,-0.38268343,0.92387953),
line(-0.38268343,0.92387953,-0.92387953,0.38268343),
line(-0.92387953,0.38268343,-0.92387953,-0.38268343),
line(-0.92387953,-0.38268343,-0.38268343,-0.92387953),
line(-0.38268343,-0.92387953,0.38268343,-0.92387953),
line(0.38268343,-0.92387953,0.92387953,-0.38268343),
line(0.92387953,-0.38268343,0.92387953,0.38268343),
locate(-.07,-.2,"45°"),
locate(-.3,-.4,10),locate(.2,-.4,10),

line(0.38268343,-0.92387953,0,0),
line(-0.38268343,-0.92387953,0,0)
 )}}}

The central angle is {{{"360°"/8 = "45°"}}}

The SAS formula for the area of a triangle is

"The area of a triangle is equal to one-half the product of
two sides multipled by the sine of the included angle.

Area of the triangle = {{{expr(1/2)(10)(10)sin("45°")}}}{{{""=""}}}{{{(50)sqrt(2)/2}}}{{{""=""}}}{{{25sqrt(2)}}}

and as we see below:

{{{drawing(400,400,-1.2,1.2,-1.2,1.2,
line(0.92387953,0.38268343,0.38268343,0.92387953),
line(0.38268343,0.92387953,-0.38268343,0.92387953),
line(-0.38268343,0.92387953,-0.92387953,0.38268343),
line(-0.92387953,0.38268343,-0.92387953,-0.38268343),
line(-0.92387953,-0.38268343,-0.38268343,-0.92387953),
line(-0.38268343,-0.92387953,0.38268343,-0.92387953),
line(0.38268343,-0.92387953,0.92387953,-0.38268343),
line(0.92387953,-0.38268343,0.92387953,0.38268343),
locate(-.07,-.2,"45°"),
locate(-.3,-.4,10),locate(.2,-.4,10),

line(0.92387953,0.38268343,0,0),
line(0.38268343,0.92387953,0,0),
line(-0.38268343,0.92387953,0,0),
line(-0.92387953,0.38268343,0,0),
line(-0.92387953,-0.38268343,0,0),
line(-0.38268343,-0.92387953,0,0),
line(0.38268343,-0.92387953,0,0),
line(0.92387953,-0.38268343,0,0)

 )}}}

the regular octagon is composed of 8 such triangles,
all congruent.  Therefore:

The area of the regular octagon = {{{8*25sqrt(2)}}}{{{""=""}}}{{{200sqrt(2)}}}.

That's approximately 282.8427125 square units.

Edwin</pre>