General Area FormulaDate: 02/14/2002 at 22:46:39 From: Will Voorhees Subject: General Area Formula I heard that there was an all-inclusive formula for the area of a square, rectangle, parallelogram, trapezoid, and triangle. Is this true? I have looked everywhere, but I can't find anything. Date: 02/15/2002 at 11:01:29 From: Doctor Ian Subject: Re: General Area Formula Hi Will, Let's start with the most complicated (i.e., least symmetric) shape, which is a trapezoid: a ________ / \ area = height * (a+b)/2 /__________\ b In the case where a and b are equal, we have a parallelogram: a ___________ / / area = height * (a+b)/2 /__________/ b And the formula still works. Note that when a = b, (a+b)/2 = (b+b)/2 = 2*b/2 = b If we make all the angles square, we have a rectangle: a _________ | | area = height * (a+b)/2 |_________| b And the formula still works. If we make the width the same as the height, we have a square: a (= h) ___ | | area = height * (a+b)/2 |___| b And the formula still works! The principal difference is that when you have a rectangle or a square, the height is trivial to find; while when you have a trapezoid or a parallelogram, the process can be somewhat more involved. So, what about a triangle? Well, if we draw the triangle so that the base is horizontal, a a /\ /| a / \ / | . / / \ / | . / /______\ /___| _____/ b b b then the value of the 'top base', a, is zero, so the formula gives us area = height * (a+b)/2 = height * (0+b)/2 = (1/2) height * b So it works for a triangle, too - if you're willing to define a triangle as a quadrilateral with one zero-length side. I hope this helps. Thanks for an interesting question! - Doctor Ian, The Math Forum http://mathforum.org/dr.math/ Date: 02/15/2002 at 22:46:24 From: Will Voorhees Subject: General Area Formula Thanks for the reply! This really helped me a lot. William Voorhees |
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