Prove Heron's Formula (Trigonometry)

In summary, the conversation discusses the proof of Heron's Formula, which calculates the area of a triangle using its side lengths. The formula involves the semiperimeter of the triangle and the sine function. The conversation also includes a hint to use the area formula with the sine function. One of the participants shares their approach, while the other struggles to understand.
  • #1
rocomath
1,755
1

Homework Statement



If a, b, c are the lengths of the sides of a triangle, then the area K of the triangle is given by [tex]K=\sqrt{s(s-a)(s-b)(s-c)}[/tex], where [tex]s=\frac{1}{2}(a+b+c)[/tex]. The number s is called the semiperimeter. Prove Heron's Formula. Hint: Use the area formula [tex]K=\frac{1}{2}bc\sin\phi[/tex].

sinphi should be sinA. it wouldn't let me use sinA.

The Attempt at a Solution



Absolute torture if you ask me! I need help getting on the right track, any help is appreciated.

Ignore everything from the triangle and down, that's a different problem.

http://img206.imageshack.us/img206/697/53149485pi4.jpg [Broken]​
 
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  • #2
my way:
K=whatever
K^2=whatever without sqrt

K^2 = that sin theta area^2

and using identidy sin^2 = 1-cos^2 in above

and cosine law, and some simplication, you would eventually reach somewhere like
-a^4+6a^3+3a^2...

and now just expand that herione thing

lol >(evil smile)<
 
  • #3
rootX said:
my way:
K=whatever
K^2=whatever without sqrt

K^2 = that sin theta area^2

and using identidy sin^2 = 1-cos^2 in above

and cosine law, and some simplication, you would eventually reach somewhere like
-a^4+6a^3+3a^2...

and now just expand that herione thing

lol >(evil smile)<
lol ... i don't follow!
 
  • #4
you didn't square the equation in the starting, but I did..
 
  • #5
rootX said:
you didn't square the equation in the starting, but I did..
ok let me try it that way.
 

1. What is Heron's Formula and what does it prove?

Heron's Formula, also known as the Hero's Formula, is a mathematical formula that calculates the area of a triangle using only the lengths of its sides. It proves that the area of a triangle can be determined without needing the height or base length.

2. How is Heron's Formula derived?

Heron's Formula can be derived from the Law of Cosines, which relates the side lengths of a triangle to the cosine of one of its angles. By using this law and some algebraic manipulation, the formula can be obtained.

3. What are the applications of Heron's Formula?

Heron's Formula is commonly used in geometry and trigonometry problems to calculate the area of a triangle. It is also used in engineering, physics, and architecture to determine the surface area of irregular shapes.

4. How accurate is Heron's Formula?

Heron's Formula is a very accurate method for calculating the area of a triangle. It is exact when the sides of the triangle are known exactly, but may have some error when rounded values are used.

5. Can Heron's Formula be used for all types of triangles?

Yes, Heron's Formula can be used for all types of triangles, including acute, right, obtuse, and even equilateral triangles. However, it may not be the most efficient method for calculating the area of certain types of triangles, such as right triangles where the base and height can be easily determined.

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