Alternative way to calculate the area of a right angled triangle

In summary, the conversation involves a new member introducing themselves and discussing their interest in proofs. They share their own proof for finding the area of a right angled triangle and verify it through comparison and online research. They then ask if anyone has seen this proof before and discuss its potential usefulness. The conversation also mentions a related interesting observation about the maximum area of a triangle.
  • #1
kenneth_G
1
0
Hello all!

New to the forums, and I have a question for you. In my classes, we have been dealing a lot with proofs lately, so when I was working on an assignment, I figured I would try and find my own proof for something, just for the hell of it. I decided to tacle the area of a right angled triangle, because I was using it in one of my assignments, and I wanted to skip some of the legwork I had to do to find the area. On my third try of finding a proof, I finally arrived at something useful.

triangle02.gif

Starting with Area=½ab, I ended up with:
Area=¼ [itex]c^{2}[/itex]sin(2A)=¼ [itex]c^{2}[/itex]cos(2B)​

I verified my findings by comparing the results with the ½ab version, and they match. I also wanted to doube check it with some trusted source (and that I wasn't just getting lucky), so I searched around on the web, but can't find it listed anywhere.

So now the question: Has anyone seen this before? And where?
 
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  • #2
welcome to pf!

hello kenneth! welcome to pf! :smile:

(actually, it's ¼ [itex]c^{2}[/itex]sin(2A)=¼ [itex]c^{2}[/itex]sin(2B) :wink:)

i don't think it has a name, since it's not particularly useful

but if you double the triangle, by adding a reflection to itself, making a triangle with angles 2A B and B, then it's the well-known formula for the area of a triangle, 1/2 side*side*sin(anglebetween) = 1/2 c*c*sin2A :smile:
 
  • #3
Your result shows one (slightly) interesting thing. For a fixed length of c, the maximum area is when sin(2A) = 1, or A = B = 45 degrees.

You can see that a different way, if you know that the right angle always lies on the circle with c as the diameter. If you use the formula "base x height / 2" with c as the base, so you can see where the right angle must be to get the maximum area.

Don't worry about the fact that we don't find this incredibly exciting, or useful. You will learn a lot more by getting the habit of "playing around" with math, rather than just "remembering the right formula" to solve problems.
 

1. What is an alternative way to calculate the area of a right angled triangle?

The alternative way to calculate the area of a right angled triangle is by using the formula A = (1/2) * b * h, where b is the base of the triangle and h is the height.

2. How is this formula different from the traditional formula for the area of a right angled triangle?

The traditional formula for the area of a right angled triangle is A = (1/2) * b * a, where b is the base of the triangle and a is the perpendicular height. The alternative formula uses the same base but calculates the height differently.

3. What is the advantage of using the alternative formula?

The advantage of using the alternative formula is that it can be used to calculate the area of a right angled triangle even when the perpendicular height is not known. This makes it a more versatile formula that can be applied in different situations.

4. Can the alternative formula be applied to all types of right angled triangles?

Yes, the alternative formula can be applied to all types of right angled triangles, as long as the base and height (or perpendicular height) are known.

5. Are there any limitations to using the alternative formula?

The alternative formula is only applicable to right angled triangles. It cannot be used for calculating the area of other types of triangles, such as equilateral or scalene triangles.

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