Different Triangles With the Same Base and HeightDate: 04/16/2004 at 00:14:45 From: Maria Subject: triangles with same base and height that have the same area Why do two differently shaped triangles with the same base and height have the same area? I have been trying to figure this out by sketching it in different ways. I can see it but I am having difficulty explaining it. Date: 04/16/2004 at 12:56:34 From: Doctor Peterson Subject: Re: triangles with same base and height that have the same area Hi, Maria. I'm not sure what kind of answer you are looking for, since you say you "see it" (meaning that you believe by observation that it's true?), but can't "explain it" (meaning a proof, or just an intuitive sense of "why" or "how"?). I'll show a couple ways to see how and why it works. You know the formula for the area of a triangle, A = bh/2 which implies that the area depends only on the base and the height, so what you say HAS to be true, or the formula would be wrong. Have you seen a proof of the formula? Then that proves the claim you are questioning! But you might like a more direct demonstration. First, if you compare two triangles with the same base and height, like these, + - - - - - - - - - - - - - - - - - - - - - - -+ + / \ / / | / \ / / | / \ / / |h / \ / / | / \ / / | +-----------------+ +-----------------+ + b b you can imagine sliding each particle of one horizontally to get it in the position of the other. Each horizontal row is the same length in each triangle; so if you cut the left one into thin slices and shift them over, you will get something like the right one. The thinner the slices, the closer it will be: + - - - - - - - - - - - - - - - - - - - - - - -+ + +--+ +--+ | +-----+ +-----+ | +--------+ +--------+ |h +-----------+ +-----------+ | +--------------+ +--------------+ | +-----------------+ +-----------------+ + b b That approach is what calculus uses, with some more careful thinking to make it rigorously true. See if you can work out some of the details. This idea, taken into three dimensions, is called Cavalieri's Theorem. This page shows the idea: Area of a Parallelogram http://mathforum.org/library/drmath/view/57790.html Another approach, a little more purely geometrical, is that you can actually cut one triangle apart into pieces and reassemble them to make the second triangle. It's easier to see how you can put TWO of the left hand triangle together to make a parallelogram, then reassemble that into a rectangle, then turn that into a different parallelogram that consists of two copies of the right hand triangle. I'll show it by turning both triangles into the same rectangle. We add a second copy of each triangle: +-----------------+ +-----------------+ / \ / / / / / \ / / / / / \ / / / / / \ / / / / / \ / / / / +-----------------+ +-----------------+ Now we cut the parallelograms vertically: +-----------------+ +-----------------+ /| / / | / / | / / | / / | / / | / / | / / | / / | / / | / +-----+-----------+ +-----------+-----+ Now we move the left pieces to the right: +-----------------+ +-----------------+ | /| | / | | / | | / | | / | | / | | / | | / | | / | | / | +-----------+-----+ +-----+-----------+ So their areas are clearly the same. You may also be interested in Euclid's proof; see propositions 35-38 on this page: Euclid's Elements, Book 1 http://aleph0.clarku.edu/~djoyce/java/elements/bookI/bookI.html If you have any further questions, feel free to write back. - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/ Date: 04/17/2004 at 19:12:23 From: Maria Subject: Thank you (triangles with same base and height that have the same area) Dr. Peterson, Thank you very much! Your visual explanation helped me to see how two triangles with the same base and height must have the same area and be able to explain it myself. Now I understand WHY! |
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