Heron’s Formula: Area of a Triangle Knowing Lengths of 3 Sides: Algebraic Proof

In this video I take a break from my epic #AntiGravity Part 6 video which I have been working months on to instead go over a truly amazing formula for determining the Area of a Triangle with knowing only the lengths of the 3 sides. This famous formula is known as Heron’s Formula after the Greek Mathematician Hero (or Heron) of Alexandria as first documented in 60 AD. The formula is stated as such: The Area of a Triangle is the square root of s(s – a)(s – b)(s – c) where a, b, and c are the lengths of the 3 sides of a triangle, and s is the semiperimiter or half the perimeter of the triangle and is equal to (a + b + c)/2. Notice how this formula doesn’t require the height of the triangle or any angle in between the sides. I actually had never encountered this formula before, so I just had to make this video as soon as I learned of it! The derivation method that I used to prove the formula is by using an Algebraic method using the Pythagorean Theorem. Splitting the triangle into two smaller triangles by defining a height as the connecting side, I show that applying the Pythagorean Theorem twice and combined with a LOT of algebra, I can determine the height in terms of the lengths of the sides. Then plugging this height into the common Area = Base x Height / 2 formula, and some reformulation, we get the famous Heron’s Formula! Make sure to watch this video as it is a truly amazing example of how careful and tedious mathematics can eventually lead to a very simple and elegant final formula! #TrulyAmazingStuff

And now back to my researching for my #AntiGravity Part 6 video which is shaping up to be one of the most epic and important videos ever made in the history of what we like to call “reality”…. #STAYTUNED! #MyPhD

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[mes]

I don’t always take a break from my Epic #AntiGravity Research but when I do I usually derive Heron’s Formula for the Area of a Triangle with knowing just the lengths of the 3 sides ;)

[heliocentric]

*Question:* At 7:30 What if the height that you drew in the proof was exterior to the triangle?
*Answer:* Always choose the height relative to the longest side, and the proof will work.

[SOBIESKI_freedom]

Excellent! I have often wondered about how such an unwieldy formula is derived... and now I know. 😁😁😁😁👌👍 THANKS MR EASY! You are our Heron (not the bird 😁😊).

[siragan]

why don't you talk like normal people...you are not reciting poetry...are you???