A geometry problem

Hi  Gibsonj338

If the area of a regular octagon is equal to 500 feet, what is the length of all the sides,  the radius, and the apothem.  Please show how you got your answers and round your answers to the nearest hundreths

I assume that you mean the area is 500 feet^2    :)

Let the length of the apothem be 'h'  and the length of half a side be 'b'

\(500=16*0.5*bh\\ 500=8bh\\~\\ \mbox{Now I need to find the angle (theta) between the apothem and the side }\\ \theta=180(8-2)\div 8\div2\\ \theta=\frac{6*180}{16} \\ \theta=67.5 degrees \\\)

\(tan 67.5 = \frac{h}{b}\\ b = \frac{h}{tan 67.5}\\\)

Now solve the 2 equations simultaneously to get your solution.

\(500=\frac{8h^2}{tan67.5}\\ \frac{500\;tan\;67.5}{8}=h^2\\ h=\sqrt{\frac{500\;tan\;67.5}{8}}\\\)

sqrt(500*tan(67.5*pi/180)/8 = 12.2836618175653346

5.0880593204403051371 = 5.088059320440305(12.2836618175653346/tan((67.5*pi/180))*2 = 10.17611864088061027431371e0

the apothem is  12.28366 feet and the   side is 10.176118 feet

the apothem is  12.28 feet and the   side is 10.18 feet       to the nearest hundreth foot

check

8*(0.5*12.28*10.18) should equal 500

8*(0.5*12.28*10.18) = 500.0416    good

so that seems ok I think.

Nice work, Melody.....I have a little different approc\ach......but...it should still work out the same

The total area can be found by the trig "formula"

500  = 8*(1/2)(radius)^2 *sin (45)

500 = 4 * (radius)^2 * [1 / √2]

500 = (4/√2) * (radius)^2

125√2  = (radius)^2   take the square root of both sides

√ [125√2]  = r ≈  13.2957  ft   ≈  13.3 ft [rounded]

And using the Law of Sines, (1/2) the side length, s, is given by

(1/2)s / sin (22.5)  = √ [125√2] / sin(90)

s = 2*sin(22.5) *√ [125√2]   ≈    10.176 ft  ≈ 10.18  ft [rounded]

And using the Law of Sines, again, we can find the apothem, a, as :

a / sin(67.5)  =  √ [125√2] / sin(90)

a = sin(67.5)* √ [125√2]  ≈ 12.28 ft