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. A smaller hexagon is inside of a larger hexagon. The radius of the larger hexagon is 2 feet and the sides lengths of 1. 52 feet. 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 1

Regular octagon has 8 sides, all of same length. The approximate area of the octagonal window specified, along with the frame is: Option D: 11.2 ft²

How to find the area of a regular polygon whose radius from center to vertex is given?

We can cut that polygon in pieces such that slices are from center to vertices. Then we will get isosceles triangles (n such triangles, n being the number of sides in that polygon).

Area of polygon = n times area of such isosceles triangles (all are congruent, so same area).

What is the area of an isosceles triangle?

An isosceles triangle has two of its side length of same measure.

Let the considered isosceles triangle has two same lengthed sides of 'a' units.

Let its third side be of 'b' units.

Then, if we drop a perpendicular from the vertex joining similar sides to the opposite side, the opposite side is cut in half, each of length b/2 units.

Using the Pythagoras theorem, we get the length of that perpendicular as:

$L = \sqrt{a^2 - (\dfrac{b}{2})^2} = \dfrac{\sqrt{4a^2 - b^2}}{2} \: \rm units.$

Thus, area of that triangle = 1/2 times base (b) times height(L)

Thus, $A = \dfrac{1}{2} \times b \times \dfrac{\sqrt{4a^2 - b^2}}{2} = \dfrac{b\sqrt{4a^2 - b^2}}{4} \: \rm unit^2$

Refer to the figure attached below.

The area of the window = $8 \times A$

where A is the area of each of the triangle pieces.

The length of the similar side = a = 2 ft.

The length of the third side = b = 1.52 ft.

Thus, the area A of that isosceles triangle is found to be

$A = \dfrac{b\sqrt{4a^2 - b^2}}{4} = \dfrac{1.52 \times \sqrt{4.(2^2) - (1.52)^2}}{4}\\\\A = \dfrac{1.52 \times \sqrt{13.6896}}{4} \approx 1.4059 \: \rm ft^2$

Thus, area of window = 8 times A ≈ 11.247  ≈ 11.2 ft²

Thus, The approximate area of the octagonal window specified, along with the frame is: Option D: 11.2 ft²

Learn more about area of octagon here:

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Answer 2

Answer:

11.2 ft2.

Step-by-step explanation: