Question

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

Explanation:

Step1: Calculate the area of one triangle

The formula for the area of a triangle is \(A=\frac{1}{2}bh\). For a regular octagon with radius \(r = 2\) ft and side - length \(s=1.52\) ft. The central angle of each of the 8 isosceles triangles that make up the octagon is \(\theta=\frac{360^{\circ}}{8} = 45^{\circ}\).
We can also use the formula \(A=\frac{1}{2}r^{2}\sin\theta\) (where \(r\) is the radius of the circum - circle of the regular polygon and \(\theta\) is the central angle). Here, \(r = 2\) ft and \(\theta = 45^{\circ}\), and \(\sin45^{\circ}=\frac{\sqrt{2}}{2}\approx0.707\).
\(A_{1}=\frac{1}{2}\times2^{2}\times\sin45^{\circ}\)
\(A_{1}=\frac{1}{2}\times4\times0.707 = 1.414\) \(ft^{2}\)

Step2: Calculate the area of the octagon

Since the octagon is composed of \(n = 8\) congruent isosceles triangles.
The area of the octagon \(A=n\times A_{1}\)
Substitute \(n = 8\) and \(A_{1}=1.414\) \(ft^{2}\) into the formula.
\(A = 8\times1.414=11.312\approx11.2\) \(ft^{2}\)

Answer:

\(11.2\ ft^{2}\)