Area of an Octagonal HouseDate: 08/24/2003 at 01:24:38 From: Jim Subject: Octagon house square footage I plan to build an octagonal home, and would like to know the area of the completed home if each wall span is 14 feet in length. Any information you could offer on this subject would be appreciated very much. Thank you! Date: 08/24/2003 at 04:37:32 From: Doctor Jeremiah Subject: Re: Octagon house square footage Hi Jim, An octagon is eight triangular pieces: +------L------+ + \ / + + \ / + + \ / + + \ / + | + \ a / + | | + \ / + | | + | | + / \ + | | + / \ + | + / \ + + / \ + + / \ + + / \ + +-------------+ Each triangle has a side length L and an inside angle a. Here is one of the triangles: +------L------+ --- \ / | \ / | x x | H \ / | \ a / | \ / | + --- We know that a = 360/8 and we also know that L = 14 feet. The area of this triangle is 1/8 of the area of the whole place. Any time an angle is involved usually trigonometry is needed to find an answer. This problem is no exception. In this case we need the trigonometric tangent function. Trigonometry is much easier to use with triangles that have a 90-degree angle. If we break our triangle in half it will have a 90-degree angle: + - - - +---L/2-+ |90 / \ | / H / \ | / | / \ | <---- angle = a/2 |/ + In this case the tangent of the angle (a/2) is equal to the opposite side (L/2) divided by the the adjacent-side (H), or written mathematically: tan(a/2) = (L/2)/H We know the value of a is 360/8 so: tan((360/8)/2) = (L/2)/H tan(180/8) = (L/2)/H And then we solve for H: H = L/(2 tan(180/8)) H = L/(2 tan(22.5)) Now, remember that we said that the area of the big-triangle was the length of the base (L) times time-the height (H) divided by 2? Well, we can solve this-by starting with: Triangle_Area = (LH)/2 We know the area is 3000/8 and we know something about H so we can substitute those in: Triangle_Area = (L(L/(2 tan(22.5))))/2 Triangle_Area = (L L)/(4 tan(22.5)) Triangle_Area = (L^2)/(4 tan(22.5)) Now, we know that L = 14 so we can say: Triangle_Area = (14^2)/(4 tan(22.5)) Triangle_Area = 118.3 square feet Now this is the area of each triangle and since there are 8 of them the entire place has an area of Octagon_Area = 946.37 square feet Here is a bit more info. If we can start with a square: + -- D -- +-------L-------+ -- D -- + / \ | / \ | D L L D | / \ | / \ + + | | | | | | L L | | | | | | + + \ / | \ / | D L L D | \ / | \ / + -- D -- +-------b-------+ -- D -- + Notice the triangles in the corners. Because they have a 45-degree angle the amount labeled "D" cut off each side of the board is the same. But also the long side of the triangle and the uncut part of the board must be the same. The Pythagorean Theorem says that the sum of the squares of the two short sides equals the square of the long side. So, using ^2 to mean squared, it basically says D^2 + D^2 = L^2, which is the same as 2D^2 = L^2 Now we know that the length of L is 14 feet but since 2D^2 = L^2 then D = 9.9 feet so the area cut off each corner is 49 square feet. If this were the whole square the square footage would be 1142.37 square feet. Since the octagon part is 946.37 square feet and the full square would be 1142.37 square feet the corners that are cut off account for 17.2% of the area of the full square. The octagon accounts for 82.8% of the total area. - Doctor Jeremiah, The Math Forum http://mathforum.org/dr.math/ Date: 08/24/2003 at 05:46:41 From: Jim Subject: Thank you (Octagon house square footage) Thank you very much for your quick response, and for the excellent information you gave me. I appreciate it greatly! |
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