Area of a Rhombus and Area of a Square
Date: 11/16/2007 at 15:01:21 From: Linda Subject: area of a rhombus vs. area of a square If a rhombus is a figure with four equal sides, isn't that the same as a square "pushed over"? Why do we have to use base times perpendicular height to find the area, rather than side times side as in a square? If I calculate the area of a 4cm square, it is 4x4. If I pushed this square over a bit, wouldn't I then have a rhombus? And then the area would have to remain 4x4. But I know the area formula for a rhombus says otherwise. I get that there is a difference between the slant height and the perpendicular height, but wouldn't the two areas still have to be the same? I'm sure there's a simple explanation, but I can't find it. Thanks!
Date: 11/16/2007 at 15:33:15 From: Doctor Peterson Subject: Re: area of a rhombus vs. area of a square Hi, Linda. As you push the square over, you are keeping the (now slanted) side the same length, so it is a rhombus; the height is reduced, right? Now push it farther and farther until the top reaches the ground-- it's now a flat line rather than a rhombus. Its area is now zero, right? Can you see that as you kept pushing, the area must have been getting smaller all the time, and not staying the same? Here's another way to see it. This is a rhombus: +----------+ / / / / / / / / +----------+ We can cut off a right triangle on the left and paste it onto the right to make a rectangle: +----------+ +----------+ /: /: | | / : / : --> | | / : / : | | / : / : | | +----+-----+....+ +----------+ But the height of this rectangle is less than the height of the original square, so its area is less. In fact, the area of the rhombus (or any parallelogram) is the base times the height, not the base times the other side. Ultimately, your error was in assuming that the area is determined only by the sides, and can't vary when you change the angles between them. Does that help? - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/
Date: 11/20/2007 at 13:22:44 From: Linda Subject: Thank you (area of a rhombus vs area of a square) Thanks so much! I get it now!
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