Area of a Parallelogram

Showing that the area of a parallelogram is base times height

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We know that quadrilateral ABCD over here is a parallelogram.

And what I want to discuss in this video is a general way of finding the area of a parallelogram.

In the last video we talked about a particular way of finding a area of a rhombus.

You can take half the product of it's diagonals.

And a rhombus is a parallelogram, but you can't just generally take

the product of the half the product of the diagonals of any parallelogram

It has to be a rhombus. So now were just going to talk about the parallelograms.

So what do we know about parallelograms?

Well we know the opposite sides are parallel.

That side is parallel to that side and this side is parallel to this side.

And we also know that opposite sides are congruent.

So this length is equal to this length.

And this length is equal to this length over here.

Now if we draw a diagonal.

I'll draw a diagonal. A, C

We can split our parallelogram into two triangles.

We've proven this multiple times that these two triangles are congruent.

We can do it in a pretty straightforward way.

We can look, obviously A,D is equal to B,C.

We have D,C is equal to A,B.

And then both of these triangles share this third side right over here.

They both share A,C.

So we can say triangle, I'll write this in yellow.

So we can say triangle ADC is congruent to triangle, so we want to get this right.

So it's going to be congruent to triangle, I said ADC.

So I went along this double magenta slash first, then the pink, and then I went the last one.

So I'm going to say CBA because I went the double magenta then pink then the last one.

So CBA, triangle CBA.

And this is by Side Side Side (SSS) congruency.

All three sides, they have three corresponding sides that are congruent to each other.

So the triangles are congruent to each other.

And what that tells us is that the areas of these two triangles are going to be the same.

So if I want to find the area, the area of ABCD, the whole parallelogram.

It's going to be equal to the area of triangle ADC plus the area of CBA.

But the area of CBA is the same thing as the area of ADC.

Because they are congruent by Side Side Side (SSS).

So this is just going to be two times the area of triangle ADC.

Which is convenient for us because we know how to find the areas of triangles.

The area of triangles is literally just one half times base times height.

So it's one half times base times height of this triangle.

And we are given the base of ADC.

It is this length right over here.

It is DC. You could view it as the base of the entire parallelogram.

And if you wanted to figure out the height,

we could draw an altitude down like this.

So this is perpendicular. We could call that the height right over there.

So if you want the total area of parallelogram ABCD,

It is equal to two times one half times base times height.

Well two times one half is just 1.

So you're just left with base times height.

So it's just b times this height over here. Base times height.

So thats a neat result and you might've guessed that this would be the case.

But if you want to find out the area of any parallelogram

and if you can figure out the height

it is literally, you just take one of the bases because opposite sides are equal times the height.

So thats one way you could of found the area.

Or you could've multiplied, the other way to think about it

is if I were to turn the parallelogram over, it would look something like this...

So if I were to rotate it like that.

Stand it on this side, so this would be point A

This would be point D.

This would be point C.

And this would be point B.

You could also do it this way, you could say it would be the area of this would be base times height.

So you could say h times DC.

So you could say this is going to be equal to h times the length of DC.

That's one way to do it, that's this base times this height.

Or you could say it's equal to AD times

I'll call this altitude right here h2.

Maybe I'll call this h1.

So you could take this base times this height.

Or you could take this base times this height right over here.

This is h2. Either way.

So if someone were to give you a parallelogram.

Just to make things clear.

Obviously you'd have to be able to figure out the height.

So if someone were to give you a parallelogram like this,

they were to tell you this is a parallelogram.

If they were to tell you this length right over here is 5.

And if they were to tell you that this distance is 6.

Then the area of this parallelogram would literally be 5 times 6.

I drew the altitude outside the parallelogram.

I could've drawn it right over here as well, that would also be 6.

So the area of this parallelogram would be 30.