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### 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

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/
```
Associated Topics:
Middle School Geometry
Middle School Triangles and Other Polygons

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