Answer:
ratio = 0.89
Explanation:
Let call R the radius of the circle then
area of circle is A(c) = π*R²
Now the area for a regular octagon is A(o) = 1/2 p*a where
p is the perimeter of the octagon and a the apothem
Then p = 8*L (regular octagon) where L is the side of the octagon
A(o) = 1/2* 8*L*a ⇒ A(o) = 4*L*a
We need to compute L and a
In a regular octagon, the triangles formed by two consecutive vertex and radius of the circle from these vertex, are isosceles triangle, having 45° as the internal angle
360/8 = 45°
Then as the sum of internal angles in any triangle must be 180° we have
that the others two equal angles should be
180 - 45 = 2α ⇒ α = 67,5 °
Then looking one of these triangles we see that apothem forms a right triangle, with 1/2 side of octagon and hypothenuse radius of circle) then
sin α = sin 67.5° = a / R ⇒ a = 0.923*R
and L can be calculated by
cos α = 0.38 and 0.38 =( L/2)/R
Then L = 0.38*2*R ⇒ L = 0.76*R
A(o) = 4*L*a ⇒ A(o) = 4*0.76*R*0.923*R ⇒ A(o) = 2.81*R²
Then the question what is the ratio of the area of the octagon to the circle is?
ratio = 2.81*R² / π*R²
ratio = 2.81/3.14
ratio = 0.89