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Area of Trapezoids

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Let us recall that a trapezoid is a quadrilateral that has exactly one pair of sides parallel. The area formula for trapezoids is based on the area formulas that have been previously derived and discussed. Now let us do the following steps to derive the area formula for trapezoids.

Draw a trapezoid ABCD and label its bases as $b_1$ and $b_2$ , respectively.
Fold it so that the bases meet.

Cut along this folded line, which is actually the median of the trapezoid.
Label the height of each piece with $\frac{h}{2}$ since the original height of the trapezoid is now being cut into two equal heights.
Rotate the top half part clockwise so that $b_1$ and $b_2$ are on the same line.

Therefore, the area of the trapezoid above is equal to the area of the parallelogram formed below. Using the area formula for parallelogram, we have

$A=\left(b_1+b_2 \right) \displaystyle \frac{h}{2}=\displaystyle \frac{h}{2}\left(b_1+b_2 \right)$

Area Formula for Trapezoid

$A=\left(b_1+b_2 \right) \displaystyle \frac{h}{2}=\displaystyle \frac{h}{2}\left(b_1+b_2 \right)$
where $b_1$ and $b_2$ are the bases and $h$ is the height of the trapezoid.

Examples

Find the area of each figure below.

Solutions:

1. The height of the trapezoid is unknown, so we solve for using the Pythagorean Theorem.

$\begin{align*} &h=\sqrt{3^2-2^2}=\sqrt{5}\approx 2.236\\&b_2=5\;cm+2\;cm=7\;cm\\ &b_1=4\;cm \\ \therefore &A=\displaystyle \frac{h}{2}\left(b_1+b_2 \right)=\displaystyle \frac{2.236}{2}cm\left(4\;cm+7\;cm \right)\approx 12.298\;cm^2 \end{align*}$

2. The height of the trapezoid is unknown, so we solve for $h$ using the Pythagorean Theorem. The shorter leg of the triangle formed with the altitude and the leg of the trapezoid is equal to 1 unit, that is $\left(7-5\right)\div 2$ .

$\begin{align*} &h=\sqrt{3^2-1^2}=\sqrt{8}\approx 2.828\\&b_1=7\;cm \\& b_2=5\;cm\\ &\therefore A=\displaystyle \frac{2.828}{2}cm\left(5\;cm+7\;cm \right)\approx 16.971\;cm^2 \end{align*}$

3. All the necessary parts used to solve for the area of the trapezoid are known, so the area of the trapezoid is

$A= \displaystyle \frac{h}{2} \left(b_1+b_2 \right)= \displaystyle \frac{3}{2}\left(6+8 \right)=21 \;\; square \;\; centimeters$
Practice Exercises

Find the area of each figure.

Area of Trapezoids Worksheets

Tagged as: area of a figure, circles, line, parallelograms, plane figures, quadrilaterals, rectangles, squares, trapezoids, triangles

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Area of Trapezoids

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