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## Geometry (all content)

### 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 a little confused about the fourth line of your work because I'm trying to figure out how did you get a #2 next to the "ca over 2 and turn the denominator-2 into 4"?
• its because 1/2 = 2/4, so ca/2 = 2ca/4
• 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.
• at , where did c2/4 came from?
• it is rather simple if you sit down with pen and paper to solve it!
here it goes,
c/2 = sqrt( c^2/ 4)
= (c^2/4)^1/2 ... because the power of the sort we need to find can be inverted to give the same result.
= c ^ 2(1/2) / 4 ^ 1/2
= c ^ 2/2 {which is essentially 1 !} / sqrt 4 { because being raised to 1/2 is the same as being the sort of the inverted number}
= c/2

I HOPE THIS HELPED! but the best solution and deepest understanding will only form when you do it yourself!
• What's going on with the simplification?
(1 vote)
• A lot of it has to do with algebra... you'll need to know the formulas for a^2-b^2 (which equals (a+b)(a-b)) and (a+b)^2 or (a-b)^2 which is a^2+or-2ab+b^2
• 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.
• How did u get the square root of c square by four? I understood everything except for that. Nice vedio...u explain better than my math tearcher.
• that is because c times 1/2 is the same thing as sqrtC^2/4
when you simplify that you come up with c/2.
(1 vote)
• at how did you get 4c^2?