# Area of parallelogramÂ proof

## Video transcript

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 the
area of a rhombus. You could take half the
product of its diagonals. And a rhombus is
a parallelogram. But you can't just
generally take half the product of the
diagonals of any parallelogram. It has to be a rhombus. And now we're just going to
talk about parallelograms. So what do we know
about parallelograms? Well, we know the opposite
sides are parallel. So 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 AC-- we can split our parallelogram
into two triangles. And we've proven
this multiple times. These two triangles
are congruent. But we can do it in a
pretty straightforward way. We can look. Obviously, AD is equal to BC. We have DC is equal to AB. And then both of these
triangles share this third side right over here. They both share AC. So we can say triangle-- let me
write this in yellow-- we could say triangle ADC is
congruent to triangle-- let me get this right. So it's going to be
congruent to triangle-- so I said A, D, C. So I went
along this double magenta slash first, then the
pink, and then I went D, 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 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 of
ABCD, the whole parallelogram, it's going to be equal
to the area of triangle-- let me just write it here--
it's equal to the area of ADC plus the area of CBA. But the area of CBA is just the
same thing as the area of ADC because they are congruent,
by side, side, side. 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. Area of triangles is literally
just 1/2 times base times height. So it's 1/2 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 we 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 one. And so you're just left
with base times height. So we can call this b. So it's just b times this height
over here, base times height. So that's a neat result. And you might have guessed
that this would be the case. But if you want to find the
area of any parallelogram, and if you can figure
out the height, it is literally you just
take one of the bases, because both bases are going
to be the-- opposite sides are equal, so it could have been
either that side or that side, times the height. So that's one way you
could have found the area. Or you could have multiplied. The other way to think about it
is you could have multiplied. So if I were to turn
this parallelogram over, it would look
something like this. It would look
something like this. So if I were to
rotate it like that, and stand it on this side,
so this would be point-- let me draw the points--
this would be point A. Let me make sure I'm
doing this right. Yeah. This would be
point A. This would be point D. This
would be point C. And then this would be point B.
You could also do it this way. You could say it's
1-- sorry, not 1/2. That would be for a triangle. The area of this would
be base times height. So you could say
it's h times DC. So you could say
this is going to be equal to h times the length 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, I'll call this height 2. Times h2. Maybe I'll call this h1. h1, h2. 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 have some way to be
able to figure out the height. So if someone were to give
you a parallelogram like this, they would tell you
this is a parallelogram. If they were to tell you that
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
of the parallelogram. I could have drawn it
right over here as well. That would also be 6. So the area of this
parallelogram would be 30.