An octagon is inscribed in a circle as shown above. What of the area

Area of octagon = [Area of square whose side length is 3+2√2 ] - (Area of 4 isosceles right angle triangle whose leg is √2]

Solution:

If you have trouble finding the exact area of the octagon above, you can just find an approximate area by considering it as a regular octagon with side length of 2.5 (notice that we use 2.5 since it’s the average of 2 and 3). The argument is: even though the regular octagon does not have the same area as the one shown above, it will be pretty close since a regular octagon with side length 2.5 can also be divided into 8 triangles each with base 2.5. Each of these triangles is somewhat greater than the one above that has base 2, but it is also somewhat less than than the one above that has base 3. So they “average” out about the same area. That is, a regular octagon with side length of 2.5 more or less has the same area as the one shown above.

Now we can use the following fact about the area of a regular octagon with side length of s:

Area = 2s^2 + 2s^2√2 (we will leave readers to show this is the case)

Therefore, if s = 2.5, the area would be 2(2.5)^2 + 2(2.5)^2√2 = 12.5 + 12.5√2.

We see that choice D’s 13 + 12√2 is the closest to 12.5 + 12.5√2; therefore, choice D must be the correct answer.

Answer: D