SOLUTION: A square is inscribed in a semi circle having a radius of 15m. The base of the square is on the base diameter of the semi-circle. Find the area of an octagon inscribed in the squ

I'm quite sure you mean

We want to find the dimensions of the square, given the radius 15m.
We draw a radius (in green) from the center of the semicircle to the upper
right corner of the square.  The shorter leg of the right triangle formed 
is one-half the side of the square and the vertical side is the whole side.
So we let half the side of the square be x and the whole side be 2x. The
hypotenuse is the radius of the circle which is 15.  We use the Pythagorean
theorem to find x.





Now we know that each side of the square is twice that or 6√5 meters.
So the area of the square is 

 square meters.

We will draw in the regular octagon by cutting off the 4 corners of the
square, which are 4 45-45-90 right triangles, the appropriate size. We
let each side of the regular octagon be s.  We let the legs of the 4
45-45-90 right triangles be y.



We use ratio and proportion to find y by comparing the right
triangle to the standard 45-45-90 right triangle with sides
1,1,√2.




rationalize the denominator


We know that y+s+y = side of the square, 
which we found = 6√5







rationalize:


We find y






The area of each of the right triangles is







}

}

}
}
}

Since there are 4 of them, the amount we must subtract from the
area of the square is 4 times that or

}

The area of the square is 180 square meters, so we subtract that
and get the area of the regular octagon as



That's about

 square meters.
 
Edwin