Draw straight lines from each corner of the octagon to the center of the pentagon to create 8 identical triangles. The length of the base b of each of these triangles is 6ft, because each is a side of the octagon. If you know the height h of these triangles, which is the length from the center of the octagon to the midpoint of a side and is called the apothem, you can easily calculate the area A of the octagon:
A = 8 x b x h / 2 = 48ft x h / 2.
How do you calculate the length of the apothem h? You need to calculate the values for some angles in these triangles and use trigonometry. There are 8 isosceles triangles that meet in the center of the octagon and complete a 360 degree full circuit, so each has a central angle of 360/8 = 45 degrees. Since the angles of a triangle total 180 degrees and these are isosceles triangles the two other angles = 1/2 x (180 - 45) = 62.5 degrees each. Looking at 1/2 of one of these isosceles triangles, where one side is length h and the side perpendicular to h has length 1/2 x 6ft = 3ft, you can use:
tan(62.5) = opposite/adjacent = h/3ft to find the value of h:
h = 3ft x tan(62.5) = 7.24264068712ft
So, this is the area A of an octagon with 6ft sides rounded to the nearest tenth:
A = 48ft x 7.24264068712ft / 2 = 173.8 ft2
