Area of a TrapezoidDate: 04/27/97 at 11:50:55 From: John Browne Subject: Area of an uneven rectangle The measurements of an uneven rectangle are 80' x 80' x 80' x 120'. We would like to know the area and the the acreage of this region and how you get the answer. Thanks for your help. John Date: 07/14/97 at 17:57:40 From: Doctor Terrel Subject: Re: Area of an uneven rectangle Dear John, What an interesting problem you are proposing! First, an "uneven" rectangle is a new idea for me. By definition, true rectangles always have two pairs of sides equal in length; plus the pairs of "opposite sides" are always parallel. So your 80' x 80' x 80' x 120' figure really isn't actually a rectangle. However, I've been trying to think what it might be, and the only idea that comes to me right now is a TRAPEZOID. Here's how I think your figure might be: A___________B / \ AB = BC = AD = 80 / \ / \ / \ CD = 120 D/___________________\C (Angles ADC and BCD have the same measure.) Now the area of a trapezoid is rather easy to find. We use this formula: h(b + b') A = ---------- h = height, b = one base, and b' = other base 2 In our problem, b = CD = 120 and b' = AB = 80. We just need to find the height "h". For that we need to add some more information to our drawing. A___________B /| |\ / | | \ / | | \ / | | \ D/____|_________|____\C E F Since our trapezoid is an "isosceles" one [i.e. the non-parallel sides, AD and BC, are the same length], we can draw the segments AE and BF. Either one of these can be considered the height of our figure. Since AB = EF = 80, then DE = FC = 20 and we can use the Pythagorean theorem on either "right" triangle - the one on the right (BCF) or the one on the left (AED) - to find the length of those segments. AE^2 + DE^2 = AD^2 AE^2 + 20^2 = 80^2 AE^2 + 400 = 6400 AE^2 = 6000 AE = sqrt(6000) Now substituting our numbers into the formula, we have: sqrt(6000)(80 + 120) sqrt(6000)(200) A = --------------------- = ----------------- = 100 sqrt(6000) 2 2 Using a calculator to simplify the result, we obtain 7746 sq ft (approximately). Finally, you asked for its acreage. There are 43,560 sq ft in one acre. So divide 7746 by 43560 (another good time to use your calculator!), yielding 0.1778 acre (again a rounded value). It is interesting to observe that this shape or field is a little bit more than one-sixth of an acre (1/6 = 0.1667 (approx.)). I hope this helps you. Write again. -Doctor Terrel, The Math Forum Check out our web site! http://mathforum.org/dr.math/ |
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