Another area problem

[firemeboy]

Ok, I think I have the area of a square down, and now I can figure the area of a circle as well as its volume.

What about the area of a triangle?

I know this is probably very simple for most people, but I dont remember how to do it.

[Amy]

It's bh/2, where b=the length of the base (whichever side you are calling the "bottom" of the triangle) and h=the height, or the vertical distance from the line containing the triangle's base to the angle formed by the other two sides. It's a lot easier to show than to explain in words, but I don't have time at the moment to fiddle around with ASCII graphics trying to draw it.

[Sofis]

What Amy is saying is, basically, a triangle is half a square.

[Amy]

Well, half a rectangle, anyway.

[firemeboy]

The angles of the triangle don't matter? Ok, I think I got it, why don't you give me a problem and see if I can get it.

[This message has been edited by firemeboy (edited 02-22-2000).]

[araya]

alright, I'll be professor araya today. Here's your assignment.

code:

---

Find the areas of these two triangles.





  |\                   / \


   | \                 /   \


   |  \ 10          6 /     \ 6


 8 |   \             /       \


   |    \           /         \


   |     \         /           \


   -------         -------------


      6                  6

---

[Sofis]

Stupid me. Half a rectangle, of course.

[Alfie]

[nitpick]It is actualy half a parallelogram. The area of a | |ogram is bh. So the area of a triangle is bh/2[/nitpick]

[firemeboy]

Ok, the area of the first triangle is 24. That one was simple enough. But on the second rectangle, how do you find the height? I am going to try some things and make a guess.

[firemeboy]

It isn't 18 is it?

[Amy]

Try applying the Pythagorean theorem.

Alfie--well, yes, it is half a parallelogram, but a parallelogram is essentially a rectangle in terms of its area--just part of it has been hacked off one side and stuck on the other.

[firemeboy]

Oh boy, I don't know if I remember exactly how that goes. ab+- the square root of something or other... Anybody know?

[Quailman]

No, that's not it. The Pythagorean Theorem states that for any right triangle, the sum of the squares of the lengths of the 2 shorter sides is equal to the square of the length of the hypotenuse. (a^2 + b^2=c^2) BTW, the Scarecrow got it wrong.

In this case, find divide the 6x6x6 triangle down the middle and consider either half, which has a base of 3 and a hypotenuse of 6. The height = sqrt(6^2-3^2) = sqrt(27) = approximately 5.196. That's the height of your triangle.

[firemeboy]

Ok, it finally dawned on me. I think I can do it now.

I had a student teacher for geometry in high school and she was in to the self confidence thing. Nobody learned anything but we walked out of the class feeling pretty peachy about ourselves.

It was a waste.

[araya]

Okay, well I haven't seen any correct answers for the second triangle so I'll assume that it's fairly trivial. Here's how to solve it:

code:

---

      /|\


     / | \


  6 /  |  \ 6


   /   |h  \


  /    |    \


  -----------


    3     3

---

Since pythagorus states that 3*3 + h*h = 6*6, then h*h = 36 - 9 = 27, so h = sqrt(27)

Then the area is 3 * sqrt(27) = 9 * sqrt(3) = 15.588 square units. Hope this helped somebody. It's extremely important to understand the pythagorean theorem. If you want more questions, please ask.

[Amy]

Technically, the area is half that.

[Aaron]

I think this would be a time to mention the basic 'sine rule' for triangles too.
A B C
--- = --- = ---
sin(a) Sin(b) Sin(c)

This will give angles and side lengths for any unknowns in a triangle.
As if you didn't know already.

Cya,

Az.

[Aaron]

Oops, my spacing didn't work.

A/sin(a)=B/sin(b)=C/sin(c)
Where A,B or C = length of side
and a,b or c = the opposite angle.

[firemeboy]

Whoops, when Quailman showed how to get the height, for some reason I was thinking it was solved.

[araya]

no Amy, technically the area is exactly that. I divided the base in half.

[Amy]

Oops! Sorry. Wasn't paying attention.

[CrystyB]

reply for firemeboy
(posted 02-22-2000 04:16 PM)
----------------------------
The angles of the triangle don't matter?

Well, using the definition of sin, h is equal to the length of the (either one) side with which it has a common vertex multiplied by the angle between this side and the base (on which the h "falls"). So A=a*b*sin(angle between a and b).

[godeloo]

ChristyB The trigonometrical formula for the area of a triangle is 1/2 a*c*sin b

[mathgrant]

Oops. Bump.

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"Mathematics is like the Nile, begins in minuteness, ends in magnificence." --Charles Caleb Colton