Area of a trapezoid

So right here, we have
a four-sided figure,

or a quadrilateral,
where two of the sides

are parallel to each other.

And so this, by
definition, is a trapezoid.

And what we want to do
is, given the dimensions

that they've given us, what
is the area of this trapezoid.

So let's just think through it.

So what would we get if we
multiplied this long base

6 times the height 3?

So what do we get if
we multiply 6 times 3?

Well, that would be the
area of a rectangle that

is 6 units wide
and 3 units high.

So that would give us
the area of a figure that

looked like-- let me do
it in this pink color.

The area of a figure that looked
like this would be 6 times 3.

So it would give us this
entire area right over there.

Now, the trapezoid is
clearly less than that,

but let's just go with
the thought experiment.

Now, what would happen if
we went with 2 times 3?

Well, now we'd be finding
the area of a rectangle that

has a width of 2
and a height of 3.

So you could imagine that being
this rectangle right over here.

So that is this rectangle
right over here.

So that's the 2
times 3 rectangle.

Now, it looks like the
area of the trapezoid

should be in between
these two numbers.

Maybe it should be exactly
halfway in between,

because when you look at the
area difference between the two

rectangles-- and let
me color that in.

So this is the area difference
on the left-hand side.

And this is the area difference
on the right-hand side.

If we focus on
the trapezoid, you

see that if we start with the
yellow, the smaller rectangle,

it reclaims half
of the area, half

of the difference between
the smaller rectangle

and the larger one on
the left-hand side.

It gets exactly half of
it on the left-hand side.

And it gets half the
difference between the smaller

and the larger on
the right-hand side.

So it completely makes
sense that the area

of the trapezoid, this
entire area right over here,

should really just
be the average.

It should exactly be
halfway between the areas

of the smaller rectangle
and the larger rectangle.

So let's take the average
of those two numbers.

It's going to be 6 times 3 plus
2 times 3, all of that over 2.

So when you think about
an area of a trapezoid,

you look at the two bases, the
long base and the short base.

Multiply each of those times
the height, and then you

could take the average of them.

Or you could also
think of it as this

is the same thing as 6 plus 2.

And I'm just factoring
out a 3 here.

6 plus 2 times 3, and
then all of that over 2,

which is the same
thing as-- and I'm

just writing it
in different ways.

These are all different
ways to think about it--

6 plus 2 over 2, and
then that times 3.

So you could view
it as the average

of the smaller and
larger rectangle.

So you multiply each of
the bases times the height

and then take the average.

You could view it as-- well,
let's just add up the two base

lengths, multiply that times the
height, and then divide by 2.

Or you could say, hey, let's
take the average of the two

base lengths and
multiply that by 3.

And that gives you
another interesting way

to think about it.

If you take the average of these
two lengths, 6 plus 2 over 2

is 4.

So that would be a width
that looks something

like-- let me do this in orange.

A width of 4 would look
something like this.

A width of 4 would look
something like that,

and you're multiplying
that times the height.

Well, that would be a rectangle
like this that is exactly

halfway in between
the areas of the small

and the large rectangle.

So these are all
equivalent statements.

Now let's actually
just calculate it.

So we could do any of these.

6 times 3 is 18.

This is 18 plus 6, over 2.

That is 24/2, or 12.

You could also do it this way.

6 plus 2 is 8, times 3 is
24, divided by 2 is 12.

6 plus 2 divided by 2
is 4, times 3 is 12.

Either way, the area of this
trapezoid is 12 square units.