Area of a Triangle Explained

Date: 06/17/99 at 01:03:35
From: Tracy Watson
Subject: Finding area of triangle

Dear Dr. Math,

I am a homeschool mother in need of a little help on how to explain 
how to find the value of A, when the area = 48 and B = 8. The formula 
we are using is as follows:

   A = (1/2)ab

I can look in the teacher key and see how to work the problem but I 
don't know how to explain it to him. Please help if you can!

Thanks, 
Tracy Watson

Date: 06/17/99 at 01:33:55
From: Doctor Jeremiah
Subject: Re: Finding area of triangle

Hi Tracy,

The formula for the area of a triangle is Area = 1/2 * a * b where "a" 
is the height and "b" is the width of the base.

                +        ---
               / \        |
              /   \       |
             /     \      a
            /       \     |
           /         \    |
          +-----------+  ---

          |-----b-----|

It doesn't matter whether the triangle is right angled or not (but it 
is easier to explain if it is right angled.) Here is how I would 
explain it with a right triangle:

Think about the area of a rectangle...

          +-----+  ---
          |     |   |
          |     |   |
          |     |   a
          |     |   |
          |     |   |
          +-----+  ---

          |--b--|

     RectangleArea = a * b

Now split it up into two triangles...

          +-----+  ---
          |    /|   |
          |   / |   |
          |  /  |   a
          | /   |   |
          |/    |   |
          +-----+  ---

          |--b--|

Two triangles fit into the same area so

     RectangleArea = 2 * TriangleArea

And then if we combine these two equations...

        RectangleArea = a * b  <==  RectangleArea = 2 * TriangleArea

     2 * TriangleArea = a * b

Then multiply both sides by 1/2 ...

           2 * TriangleArea = a * b
     1/2 * 2 * TriangleArea = 1/2 * a * b
               TriangleArea = 1/2 * a * b

And there we have the equation for the area of a triangle.


To explain triangles that do not have a right angle:

                +        ---
               / \        |
              /   \       |
             /     \      a
            /       \     |
           /         \    |
          +-----------+  ---

          |-----b-----|

First split the triangle up into two right triangles...

                +        ---
               /|\        |
              / | \       |
             /  |  \      a
            /   |   \     |
           /    |    \    |
          +-----+-----+  ---
          |--c--|
          |-----b-----|

The Area of the triangle on the left is

     LeftTriangleArea = 1/2 * a * c

The Area of the triangle on the right is

     RightTriangleArea = 1/2 * a * (b-c)

The area of the whole thing is

     TriangleArea = LeftTriangleArea + RightTriangleArea
     TriangleArea = (1/2 * a * c) + (1/2 * a * (b-c))

Now factor out the "1/2 * a" ...

     TriangleArea = (1/2 * a * c) + (1/2 * a * (b-c))
     TriangleArea = 1/2 * a * (c) + 1/2 * a * (b-c)
     TriangleArea = 1/2 * a * (c + b-c)

And simplify...

     TriangleArea = 1/2 * a * (c + b-c)
     TriangleArea = 1/2 * a * b


Let me know if this helps.

- Doctor Jeremiah, The Math Forum
  http://mathforum.org/dr.math/