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

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of the triangle if we know the
length of its base

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and its height.

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

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

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

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

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times the height.

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

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

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

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

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

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When you aren't
given the height.

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

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where I just give you the
lengths of the sides.

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

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the lengths of these sides.

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How do you figure that out?

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And to do that we're
going to apply something

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called Heron's Formula.

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

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

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And really to prove it you
already probably have

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the tools necessary.

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It's really just the
Pythagorean theorem and

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a lot of hairy algebra.

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

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

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pretty simple and pretty
easy to remember.

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

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

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S, which is essentially
the perimeter of this

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triangle divided by 2.

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a plus b plus c, divided by 2.

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

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

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

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

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That's Heron's
Formula right there.

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This combination.

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Let me square it off for you.

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So that right there
is Heron's Formula.

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

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

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times height.

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

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see this is actually
not so bad.

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So let's say I have a triangle.

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I'll leave the
formula up there.

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

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of length 9, 11, and 16.

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So let's apply Heron's Formula.

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

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So 9 plus 11 plus
16, divided by 2.

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

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36, divided by 2 is 18.

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

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

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

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And then this is equal to
the square root of 18

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times 9 times 7 times 2.

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

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So I'll just
rearrange it a bit.

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

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

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root of 9 times the
square root of 7.

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The square root
of 36 is just 6.

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This is just 3.

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

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because you can't have
negative side lengths.

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And so this is going to
be equal to 18 times

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the square root of 7.

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

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

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that, to figure out that the
area of this triangle

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right here is equal to 18
square root of seven.

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Anyway, hopefully you
found that pretty neat.

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