Question

Find the formula, in terms of K, for the area of a regular octagon with apothem K and side of 10.

A =


HELP ME!

Answer 1

This was a funion. I learn something new everyday. Moving on...

I found a tutorial that describes the process pretty well, I'll also give my explanation. http://mathcentral.uregina.ca/QQ/database/QQ.09.01/laurie2.html

What do we have? K, the apothem, is pretty much the same thing as the radius of a circle, but instead for an octagon. If we imagine for a moment that instead of working with something as daunting as an octagon, we attach 4 little small right triangles to the corners so that now we can work with a square, finding an equation for the area would be simple, since we should know that the aread of a square is one side times another side or \[a = s^2\] but with our imagined square, the apothem is only half of a side. Rather that being awkward and start working with 2's and 1/2's let us\[let, d = 2k\] I called it d because it's 2x the apothem, which is similar to the diameter of a circle, since the apothem is like the radius, and the d is 2x the apothem. We can now use our area of a square formula to get our formula so far \[a = d^2 - ...\] The only thing left to do is subtract the area of those corner triangles we added. Since I love squares so much (and the 4 triangles are right triangles of sides) we can put the triangles together to make 2 squares as well, which will be useful for finding and subtracting their area. Lets call the sides of the octagon s, and the sides of our little corners (which make up the sides of our squares), b. We need to find the area of the 2 squares (b*b) in terms of K. Looking at one side of our octagon square we notice that the side d is equal to two sides of the triangles and the side of the octagon itself\[d = b + s + b\] \[d = s + 2b\] In order to get the formula in terms of k we need to find a way to substitute s for something else. If we take one of our corners we know we have a right triangle comprised of 2 sides, b, and a hypotenuse s. We know that we can use the Pythagoras theorem to get \[s^2 = b^2 + b^2\] \[s^2 = 2b^2\] \[s = \sqrt{2b}\] So then, we've now gotten rid of s to give us\[d = \sqrt{2b} + 2b\] We're almost there, now we just need to substitute the b's so that they are in terms of d (or 2K) equating for b we get \[d = (\sqrt{2} + 2)b\] \[b = d / (\sqrt{2} + 2)\] Now that we have b we can find the area of the 2 squares (4 corners) and subtract it from our larger square to get the area of the octagon. We know that our current formula is as follows \[a = d^2 + (areaOfSquares)\] Since we now have b in terms of d (or 2K) we can find the area of the 2 squares which is simply 2*(b*b) or if we substitute from our last equation \[2 * (d / (\sqrt{2} + 2)^2\] put the whole thing together and you get \[a = d^2 - 2(d / (\sqrt{2} + 2)^2\] And we're finished! Oops, don't forget that the question wanted it in terms of K not d (or 2K) so we simply substitue 2K for d to get \[a = (2K)^2 - 2(2K / (\sqrt{2} + 2)^2\]

Answer 2

Typo \[a = d^2 + (areaOfSquares)\] is supposed to be \[a = d^2 - (areaOfSquares)\]

Answer 3

i had gotten 4500 from another person point of view

Answer 4

My equation will give you 365 with an apothem of 10... which is slightly off 331, which is what wolfram alpha gives http://www.wolframalpha.com/input/?i=area+of+octagon+of+apothem+10 I'm not sure as to where my equation's inaccuracy arises. However, I'm pretty certain that it's not 4500.

Answer 5

soit would be 331

Answer 6

Indeed, I must have typed values in my calculator wrong. My equation gives the same answer when I put it in wolfram alpha http://www.wolframalpha.com/input/?i=%282%2810%29%29^2%E2%88%922%282%2810%29%2F%28sqrt%282%29%2B2%29%29^2

Answer 7

Yey, I thought I did all of that for nothing for a minute there lol.

Answer 8

lolz

Answer 9

its not right ugh