Answer:
13 + 12√2
Step-by-step explanation:
Not sure if there's an easier way, but here's my method:
Draw the radius lines from the center of the circle to each vertex of the octagon. Â This will divide the octagon into 8 isosceles triangles, 4 big and 4 small.
Draw the height of one of the big triangles. Â Define the angle between the height and the radius as A. Â Similarly, draw the height of one of the small triangles. Â Define the angle between the height and the radius as B.
Using trig, we can say:
sin A = 1.5 / r
sin B = 1 / r
The vertex angle of the large isosceles triangle is 2A, and the vertex angle of the small isosceles triangle is 2B. Â Therefore:
4(2A) + 4(2B) = 360
A + B = 45
If we substitute into the first trig equation:
sin(45 − B) = 1.5 / r
sin 45 cos B − cos 45 sin B = 1.5 / r
1/√2 cos B − 1/√2 (1/r) = 1.5 / r
cos B − 1/r = 1.5√2 / r
cos B = (1 + 1.5√2) / r
If we substitute into the second trig equation:
sin(45 − A) = 1 / r
sin 45 cos A − cos 45 sin A = 1 / r
1/√2 cos A − 1/√2 (1.5/r) = 1 / r
cos A − 1.5/r = √2 / r
cos A = (1.5 + √2) / r
Using SAS area of a triangle, the area of the large triangle is:
Area = ½ r² sin(2A)
Area = ½ r² (2 sin A cos A)
Area = r² sin A cos A
Area = (r sin A) (r cos A)
Area = (1.5) (1.5 + √2)
Area = 2.25 + 1.5√2
Similarly, the area of a small triangle is:
Area = ½ r² sin(2B)
Area = ½ r² (2 sin B cos B)
Area = r² sin B cos B
Area = (r sin B) (r cos B)
Area = (1) (1 + 1.5√2)
Area = 1 + 1.5√2
So the total area is:
Area = 4(2.25 + 1.5√2) + 4(1 + 1.5√2)
Area = 9 + 6√2 + 4 + 6√2
Area = 13 + 12√2
