Answer:
Perimeter of platter is 39.84 inch.
Area of platter is [tex]119.52\:inch^{2}[/tex]
Step-by-step explanation:
Apothem is the perpendicular line from center to the side of the octagon. Given that apothem is 6 inch that is AB=6 inch and [tex] m\angle 90^{\circ}[/tex] (Refer attachment).
Assume DB = BC = x.
Since [tex] \angle DAC[/tex] is central angle, Therefore
[tex] m\angle DAC=\dfrac{360}{8}[/tex]
[tex] m\angle DAC=45^{\circ}[/tex]
Since apothem AB bisects angle DAC. So,
[tex]\angle BAC=\dfrac{ m\angle DAC }{2}[/tex]
[tex]\angle BAC=\dfrac{ 45 }{2}[/tex]
[tex]\angle BAC=22.5^{\circ}[/tex]
To find the value of BC, use trigonometry ratio for tan for [tex] \Delta ABC[/tex],
[tex] tan\:\theta=\dfrac{opposite\:side}{adjacent\:side}[/tex]
[tex] tan\left(22.5\right^{\circ})=\dfrac{BC}{AB}[/tex]
[tex] tan\left(22.5\right^{\circ})=\dfrac{x}{6}[/tex]
Solving for x multiply whole equation by 6.
[tex] tan\left(22.5\right^{\circ})=\dfrac{x}{6}[/tex]
[tex] 6\times tan\left(22.5\right^{\circ})=x[/tex]
[tex] x=2.49[/tex]
Now, DC = DB + BC
[tex]DC = x + x [/tex]
[tex]DC = 2 x [/tex]
[tex]DC = 4.98 [/tex]
Therefore, side of octagon is 4.98 inch .
Perimeter of octagon is,
[tex]Perimeter=number\:of\:sides\times length[/tex]
Since octagon has 8 sides, so number of side is 8
[tex]Perimeter=8\times 4.98[/tex]
[tex]Perimeter=39.84[/tex]
Therefore, perimeter of platter is 39.84 inch.
Area of octagon is,
[tex] Area\:of\:octagon=\dfrac{1}{2}\times apothem\times perimeter[/tex]
[tex] Area\:of\:octagon=\dfrac{1}{2}\times 6\times 39.84[/tex]
[tex] Area\:of\:octagon=119.52[/tex]
Therefore, area of platter is [tex]119.52\:inch^{2}[/tex]
Note: Values are calculated using two decimal places)
