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The regular octagon in the ceiling of this cathedral has a radius of 10.5 feet and a perimeter of 64 feet. What is the length of the apothem of the octagon? Round your answer to the nearest tenth of a foot.

Respuesta :

meerkat18
The apothem of a polygon is a line made from the center to the midpoint of any side of the polygon. The radius, on the other hand, is the line from the center to any corner of the polygon. If you illustrate the problem, you can form a right triangle where the hypotenuse is the radius, and the other two legs are the apothem and half of one side of the polygon. 

Perimeter of octagon = 8*length of each side = 64
length of each side = 8 ft

Therefore,
Apothem = √(10.5^2 - (8/2)^2) = 9.71 ft

Correct response:

  • The apothem of the regular polygon is approximately 9.7 feet

Methods used for finding the apothem of the octagon

The shape of the given figure = A regular hexagon

Radius of the hexagon = 10.5 feet

Perimeter of the hexagon = 64 feet

Required:

The length of the apothem, a, and the side length, s, of a regular

polygon are related by the following formula;

  • [tex]a = \mathbf{ \dfrac{s}{2 \times tan\left(\dfrac{180^{\circ}}{n} }\right) } [/tex]

The perimeter of a regular octagon, P = 8 × s

Which gives;

[tex]s = \dfrac{64}{8} = 8[/tex]

The length of a side of the hexagon, s = 8 feet

[tex]a = \mathbf{\dfrac{8}{2 \times tan\left(\dfrac{180^{\circ}}{8} \right)}} =\dfrac{8}{2 \times tan\left(22.5^{\circ} \right)} =\dfrac{4}{\left(\sqrt{2} - 1 \right) } = 4 + 4\cdot \sqrt{2} \approx 9.7[/tex]

a = 4 + 4·√2

  • By rounding to the nearest tenth of a foot, the apothem of the regular octagon, a = 4 + 4·√2 feet ≈ 9.7 feet

Learn more about regular polygons here:

https://brainly.com/question/12024089

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