Heron's Formula

Using Heron's Formula to determine the area of a triangle while only knowing the lengths of the sides

I think it's pretty common
knowledge how to find the area

of the triangle if we know the
length of its base

and its height.

So, for example, if that's my
triangle, and this length right

here-- this base-- is of length
b and the height right here is

of length h, it's pretty common
knowledge that the area of this

triangle is going to be equal
to 1/2 times the base

times the height.

So, for example, if the base
was equal to 5 and the height

was equal to 6, then our area
would be 1/2 times 5 times 6,

which is 1/2 times 30--
which is equal to 15.

Now what is less well-known is
how to figure out the area of a

triangle when you're only given
the sides of the triangle.

When you aren't
given the height.

So, for example, how do
you figure out a triangle

where I just give you the
lengths of the sides.

Let's say that's side a, side
b, and side c. a, b, and c are

the lengths of these sides.

How do you figure that out?

And to do that we're
going to apply something

called Heron's Formula.

And I'm not going to
prove it in this video.

I'm going to prove it
in a future video.

And really to prove it you
already probably have

the tools necessary.

It's really just the
Pythagorean theorem and

a lot of hairy algebra.

But I'm just going to show you
the formula now and how to

apply it, and then you'll
hopefully appreciate that it's

pretty simple and pretty
easy to remember.

And it can be a nice trick
to impress people with.

So Heron's Formula says first
figure out this third variable

S, which is essentially
the perimeter of this

triangle divided by 2.

a plus b plus c, divided by 2.

Then once you figure out S, the
area of your triangle-- of this

triangle right there-- is going
to be equal to the square root

of S-- this variable S right
here that you just calculated--

times S minus a, times S
minus b, times S minus c.

That's Heron's
Formula right there.

This combination.

Let me square it off for you.

So that right there
is Heron's Formula.

And if that looks a little bit
daunting-- it is a little bit

more daunting, clearly, than
just 1/2 times base

times height.

Let's do it with an actual
example or two, and actually

see this is actually
not so bad.

So let's say I have a triangle.

I'll leave the
formula up there.

So let's say I have a
triangle that has sides

of length 9, 11, and 16.

So let's apply Heron's Formula.

S in this situation is going to
be the perimeter divided by 2.

So 9 plus 11 plus
16, divided by 2.

Which is equal to 9 plus
11-- is 20-- plus 16 is

36, divided by 2 is 18.

And then the area by Heron's
Formula is going to be equal to

the square root of S-- 18--
times S minus a-- S minus 9.

18 minus 9, times 18 minus
11, times 18 minus 16.

And then this is equal to
the square root of 18

times 9 times 7 times 2.

Which is equal to-- let's
see, 2 times 18 is 36.

So I'll just
rearrange it a bit.

This is equal to the square
root of 36 times 9 times 7,

which is equal to the square
root of 36 times the square

root of 9 times the
square root of 7.

The square root
of 36 is just 6.

This is just 3.

And we don't deal with the
negative square roots,

because you can't have
negative side lengths.

And so this is going to
be equal to 18 times

the square root of 7.

So just like that, you saw it,
it only took a couple of

minutes to apply Heron's
Formula, or even less than

that, to figure out that the
area of this triangle

right here is equal to 18
square root of seven.

Anyway, hopefully you
found that pretty neat.