what is the proof of the hero's formula sqrts(s-a)(s-b)(s-c) and the proof of surface area of a sphere?
actually i wanted to find out about them. please help.
- acafrao341Lv 51 decade agoFavorite Answer
Here's how you prove Heron's formula. The calculations are pretty long, but straightforward.
1. The sine law implies that the area of a triangle with sides a and b subtending angle w is A = 1/2 (ab sin w); solve this for sin w.
2. The cosine law implies that, for a triangle with sides a, b, c, where a and b subtend angle w,
c2 = a2 + b2 - 2ab cos w;
solve this for cos w.
3. Now apply the Pythagorean theorem, which says that (cos w)^2 + (sin w)^2 = 1, substituting in the two expressions you just found, and crank it out. You will find
A^2 = (1/16)(a+b+c)(-a+b+c)(a-b+c)(a+b-c).
Finally, make the definition s = (1/2)(a+b+c) (s is called the semiperimeter), substitute that in, and simplify. The end result is Heron's formula.
Hope this helps,
- a_math_guyLv 51 decade ago
S.A. of sphere comes from calculus, integral of the area patch over the whole spehere. I would guess that Hero's formula could eb derived from trigonometry (law of sines and cosines) but...uggg! I'd just accept it "as is"
- openpsychyLv 61 decade ago
kindly refer to your math text book.