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### Course: Geometry (all content) > Unit 8

Lesson 9: Heron's formula# Heron's formula

Using Heron's Formula to determine the area of a triangle while only knowing the lengths of the sides. Created by Sal Khan.

## Want to join the conversation?

- why did you divide the perimeter by 2? isnt a+b+c already the perimeter?(45 votes)
- when we have to find the "s" , we have to divide the perimeter by 2(6 votes)

- At1:38, why's perimeter represented with an S and not a P?(13 votes)
- it's just a variable... variables can change... and plus, it's meaning is semi-perimeter, which means half-of-the-triangle...i.e., a+b+c / 2(2 votes)

- Why is it called heron in the first place and when was it created?(5 votes)
- It's called Heron's Formula because it is credited to the Heron (nowadays Hero) of Alexandria and proof is found in his book called Metrica written A.D. 60. A hypothesis says that Archemedes (Greek warrior) knew it over 200 years earlier and that is possible.(6 votes)

- how to find the sides when you have the area given. if the area is 4 root 5(4 votes)
- Two triangles with different side measures could have different areas. Therefore, the side measures of a triangle can not be found given only the area.(5 votes)

- does this formula work for all types of triangles(5 votes)
- Yes, the formula works for every type of triangle(3 votes)

- at1:29what does Sal mean by "nice trick"?(2 votes)
- He's referring to the formula (Heron's formula) itself.(4 votes)

- Why do we need to find S in the equation..

I know that S=a+b+c/2

Couldn't we just do that in the old way like b*h/2

Why should we use this long and difficult equation?(2 votes)- The reason we need to know what S is is because we do not know the height. This formula is actually a really easy way to solve find the area of a triangle once you know you to use it. Another was would be to divide a triangle into two right triangles. You could then use the Law of Cosines (https://www.khanacademy.org/math/trigonometry/less-basic-trigonometry/law-sines-cosines/v/law-of-cosines-example) to find the angle measures. You could then solve for the height, which would be the dividing line through the middle of the triangle, by using the trigonometric functions (sin, cos, tan, etc.). After doing all this, you could solve for the area of the triangle using A=b*h/2. However, Heron's Formula is actually much shorter and less difficult than solving it the other way. If you put some effort into learning this formula, maybe even watch Sal's video on proofing the formula, then you will find this to be a very helpful and easy to use formula.(4 votes)

- Can you do it with lengths of 10, 6 and 4? Because I seem to end up with an area of 0. Am I doing something wrong or is the formula flawed?(1 vote)
- Neither. A "triangle" with side lengths of 10, 6, and 4 would be perfectly flat and have no area. (It isn't really a triangle at all.) For three side lengths to form a triangle, the sum of the two smaller sides must be LARGER than the largest side.(5 votes)

- why can't you multiply 18*9*7* 2?(3 votes)
- I'm just going to reroute you to Johan Kanselaar's answer.

He says:

He could have. It would have been the same answer. But he's looking for intermediate numbers which he can squareroot easily. 18x2 = 36. sqrt(36) = 6. sqrt(9) = 3.

Because of the commutative property of multiplication (A*B = B*A or in this case a*b*c*d = d*a*b*c) he can swap any of the four numbers being multiplied around.

He can also do 18*9*7*2 = 18*18*7 because 9*2 = 18. sqrt(18*18*7) is like sqrt(18*18) * sqrt(7) which is 18*sqrt(7). 18*sqrt(7) = sqrt(2268)(1 vote)

- This is confusing to me. And I want to learn this early. Could someone explain this please?(3 votes)
- This formula sqrt(s(s-a)(s-b)(s-c)) is another way of solving the area of a triangle had you've not been provided the height or don't have a way of getting it currently.

The phrase "sqrt" above is square root btw.

This formula works for the most basic of triangles and works if you are provided with the sides of the triangle.

This is how you get your answer. For example, you have a triangle with sides, 7,9,11. You don't have the height. So you just sum up the sides resulting with 27. Then you need to divide the perimeter by 2 resulting with the semi-perimeter 14.5. This then sets the base for your work.

The latter is simpler. Simply do the formula I mentioned above. Square root of semi-perimeter*semi-perimeter subtract side a*semi-perimeter subtract side b*semi*perimeter subtract side c. This then results in your area of 45.754.

Hope you understand now! ;)

P.S. Note this doesn't work for all triangles so don't use this for every single triangle you need to find the area of.(1 vote)

## Video transcript

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.