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

Lesson 9: Heron's formula

# Proof of Heron's formula (2 of 2)

Sal proves Heron's Formula for finding the area of a triangle solely from its side lengths. Created by Sal Khan.

## Want to join the conversation?

• Why is the Heron's formula so important to learn?
• Competition math. Trust me, I've asked myself the same question, but it's definitely worth it
• When you perform this kind of algebra on a novel problem are there some guiding principles that lead you towards the nice reduced solutions? Or is it a lot of playing with the numbers and seeing what interesting relationships fall out in a guess/check process?
• Me too. I wish I could write to a mathematician and ask them that.
• I'm quite amazed. Not only by this proof, but with elegant proofs in general. Euler's comes to mind. Is minimalist aesthetics inherent in the nature of mathematical concepts or do we seek to make it so? That is, to make equations simpler without making it simplistic.
• Mathematical proofs are said to be beautiful if they utilize minimal theorems or assumptions and/or utilize seemingly unrelated facts for a short and succinct solution. So yes, we seek the simplest solutions. On the other hand, proofs that are long and use many related powerful theorems are said to be ugly or even clumsy. Such proofs often demonstrate our little understanding on the subject.
• How did the person, who was the first ever to derive this formula, know where he was heading?
• At about of the video, in the 3rd line of the work: when he changes the signs of (c sq. + a sq. - b sq.), doesn't he also need to change the sign of "4," which is the dominator of the fraction? Shouldn't he be multiplying by -1/-1?

Thanks for any help.
• No. Multiplying by -1/-1 is the same thing as multiplying by 1/1 (since the minuses cancel out) and that would essantially not do anything to the other fraction. -1/1 times 5/1 is the same as (-1*5)/(1*1) = -5/1 = -5. NOT however equal to -5/-1 = 5. Hope that helps.
• I wonder why mathematics is so magical... :P
• these 12 minutes felt like 12 hours lol
• if anyone is wondering, this is my explanation for why 1/2 *c is equal to sqrt(c^2/4).

1/2*c = c/2 which is equal to sqrt(c^2/2^2) which is sqrt(c^2/4)
• At , WHERE does the 2b come from? Does it just drop out of thin air?
• He changes (c+a-b)/2 to (a+b+c)/2-2b/2. This is true because:
(c+a-b)/2
c+a-b=c+a-b+b-b=a+b+c-b-b=a+b+c-2b.
(a+b+c-2b)/2
Separate the a+b+c and the -2b into separate fractions.
(a+b+c)/2-2b/2
He does a similar process with the two other fractions after that.

I hope this clarifies how the -2b appeared!
(1 vote)
• Why is 1/2*C = sqrt(c^2/4)? Is there a video explaining all of this conversions? Thanks!