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Area of Kites and Rhombuses - Concept

Teacher/Instructor Brian McCall
Brian McCall

Univ. of Wisconsin
J.D. Univ. of Wisconsin Law school

Brian was a geometry teacher through the Teach for America program and started the geometry program at his school

The area formula for a kite is found by rearranging the pieces formed by the diagonals into a rectangle. Since one side is half of a diagonal, the area of a rhombus formula is one half the product of the diagonals. An additional formula for the area of a rhombus is to use the kite formula (it works because rhombuses are technically kites). Related topics include area of parallelograms and solving formulas.

What is the area of a kite where all you know is the length of each of those diagonals? Well, let's go back to what we do know. We know that the area of a rectangle, so if I write in some right angles, label these as parallel. We know this area is going to be equal to its base times its corresponding height. So is there any way that we can take this kite and rearrange its pieces to make a rectangle.
Well, let's start off by what we know. We know that the diagonals in the kite are perpendicular. So what I'm going to do is I'm going to cut out this piece right here out of the white board and I'm going to translate it until it fits into this area right here. I'm going to do the the same thing with this piece except and actually by translate I meant translate and then rotate. So I'm going to do the same thing with this piece right here. I'm going to translate and rotate. So I'm going to redraw my kite, where we have our initial piece which is right here and we have this bottom piece right here. But then I said I translate it and rotate it. So now we have this other piece here that we cut out and up here we have another piece that we cut out. Translate it and rotate it.
So let's start off by saying well, what do we know? We know that this distance right here is the distance of diagonal 1. So I'm going to label it as d1. What else do we know about the diagonals of a kite? Well, I know that this d2 is bisected by the other diagonal. Which means this piece right here is going to be d2 divided in half.
So now we've created since we know that these are all right angles, we have created a rectangle, and we know that the area of a rectangle is the base times the corresponding height. So the area of this rectangle, our base is d2 divided by 2 times our height which is d1. So another way of writing this is to realise that d2 and d1 are in your numerator, so you just rearrange it and write it as d2 times d1 divided by 2.
The key thing to realizing this formula, was cutting out these pieces, rotating and translating and saying that the diagonals of a kite, that this diagonal is bisected by the other diagonal. But this also has to apply to a rhombus.
So there's going to be 2 ways for you to calculate the area of a rhombus. The first way is to say if you know a height and a corresponding base, then you can just calculate the area as base times height. Because a rhombus is a parallelogram.
The other way, is to say well, the diagonals of a rhombus are also perpendicular and they bisect each other. So, this formula of d1 times d2 divided by 2 has to apply.
So you've got a couple of options when you're trying to calculate the area of a rhombus. You can use a height and base or you can use d1 times d2 divided by 2.

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