Deriving Formula for Area of Trapezoid

In a previous lesson, we covered how to use the formula for the area of a trapezoid. Now, we’re going to dive into deriving the formula itself.

[latex]\large{A = {1 \over 2}\left( {{b_1} + {b_2}} \right)h}[/latex]

Let’s consider the trapezoid shown below.

We can break down the trapezoid into three sections: two triangles and a rectangle.

To calculate the area of the trapezoid, we need to sum the areas of the individual sections: triangle #1 (the yellow triangle), triangle #2 (the orange triangle), and the rectangle (the blue rectangle).

To determine the area of each shape, let’s start by labeling the relevant parts of the triangles and the rectangle. Notice that the base of triangle #1 is [latex]x[/latex], with a height of [latex]h[/latex]. Similarly, the base of triangle #2 is [latex]y[/latex], and its height is also [latex]h[/latex]. Lastly, the length of the rectangle is [latex]b_1[/latex], while its width is [latex]h[/latex], which matches the height of both triangles.

We will first calculate the area of each section:

1) Area of Triangle 1
The area of Triangle 1, with base \( x \) and height \( h \), is:

2) Area of Triangle 2

The area of Triangle 2, with base \( y \) and height \( h \), is:

3) Area of Rectangle

The area of the rectangle, with base \( b_1 \) and width \( h \), is:

\[ \text{Area of Rectangle} = b_1 \times h \]

Now, the total area of the trapezoid is the sum of the areas of the two triangles and the rectangle:

Simplifying, we get:

\[ A = \frac{xh}{2} + \frac{yh}{2} + b_1h \]

Create a common denominator by multiplying the third expression by [latex]\frac{2}{2}[/latex] so we can add them together as fractions with the same denominator of \(2\).

\begin{align*} & = \frac{xh}{2} + \frac{yh}{2} + \frac{2b_1h}{2} \\ \\ & = \frac{xh + yh + 2b_1h}{2} \\ \end{align*}

Factoring out \(\Large{\frac{h}{2}}\), we have

\[ = \frac{h}{2} \left( x + y + 2b_1 \right) \]

From the relation above (see the illustration), side \(b_2\) is the sum of sides \(x\), \(y\) and \(b_1\). Thus,

\begin{align*} & x + y + b_1 = b_2 \\ \\ & x + y = b_2\, – \,b_1 \\ \end{align*}

Substitute the value of \(x+y\), then simplify. Finally, rearrange the terms so that it looks like formula of trapezoid.

We have successfully derived the formula for the area of trapezoid!

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• Area of Trapezoid