# 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?

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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,

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