manuel wants to buy a window shade to cover the window and frame shown. the window is in the shape of a…

manuel wants to buy a window shade to cover the window and frame shown. the window is in the shape of a regular octagon. the radius of the window, including the frame, is 2 ft, and the measure of each edge of the octagonal frame is 1.52 ft. what is the approximate area of the window that needs to be covered, including the frame? 2 ft² 7 ft² 11.2 ft² 22.5 ft²

manuel wants to buy a window shade to cover the window and frame shown. the window is in the shape of a regular octagon. the radius of the window, including the frame, is 2 ft, and the measure of each edge of the octagonal frame is 1.52 ft. what is the approximate area of the window that needs to be covered, including the frame? 2 ft² 7 ft² 11.2 ft² 22.5 ft²

Answer

Explanation:

Step1: Find the central angle of each isosceles triangle

A regular octagon has (n = 8) sides. The central angle (\theta=\frac{360^{\circ}}{n}). Substituting (n = 8), we get (\theta=\frac{360^{\circ}}{8}=45^{\circ})

Step2: Find the area of one isosceles triangle

The formula for the area of a triangle is (A=\frac{1}{2}ab\sin C). For an isosceles triangle formed by two radii ((a = b=2) ft) and the side of the octagon, and (C = 45^{\circ}) (central angle). Using (A=\frac{1}{2}\times2\times2\times\sin45^{\circ}). Since (\sin45^{\circ}=\frac{\sqrt{2}}{2}\approx0.707), then (A=\frac{1}{2}\times2\times2\times0.707 = 1.414) (ft^{2})

Step3: Find the area of the octagon

The area of the octagon (A_{total}=n\times A_{triangle}). Substituting (n = 8) and (A_{triangle}=1.414) (ft^{2}) (A_{total}=8\times1.414 = 11.312\approx11.2) (ft^{2})

Answer:

(11.2\ ft^{2}) (the third option)