Trapezoid

No, "trapezoid" is not a new music genre -- it's a geometric shape! As you'll soon find out, trapezoids have all kinds of interesting properties, and we can use our knowledge of triangles and geometry to calculate their areas, perimeters, and much more. Let's take a closer look at trapezoids together:

What makes a trapezoid?

All trapezoids have four sides. But unlike squares, these sides are not necessarily equal. And unlike rectangles, all of these lines are not at right angles. However, a trapezoid does have a pair of parallel lines. The other two sides are angled but not parallel. If two corners from a parallel side have equal angles, then we have an "isosceles trapezoid" on our hands.

Each parallel line on a trapezoid is called a "base." On the other hand, the non-parallel lines are called "legs."

How to calculate the area of a trapezoid

Although this shape might seem a little strange, it's deceptively easy for us to calculate its area. Here's the formula we'll be using:

$\left(\frac{b_1+b_2}{2}\right)h=\mathrm{Area}$

"b₁" represents one base, while "b₂" represents the other. Finally, "h" represents the height or the "altitude" of the trapezoid.

For example, if a trapezoid has a height of 5 cm and two bases with lengths of 3 cm and 4 cm, then the formula would look something like this:

$\left(3+\frac{4}{2}\right)\times 5=13\text{cm}{}^2$

Finding the perimeter of a trapezoid

Finding the perimeter of a trapezoid is easy. All we need to do is add all four sides together.

For example, a trapezoid might have sides with lengths of 4 cm, 5cm, 1 cm, and 4 cm. If we add them all together, we get 14 cm.

If we have enough data, we can calculate both the perimeter and area of a trapezoid:

The perimeter is simply $8+3+10+11$ , leaving us with 32.

The area is $\frac{3+11}{2}\times 7$ , leaving us with 49.

Finding the median of a trapezoid

Ready to get really fancy with trapezoids? It's possible to calculate the median of a trapezoid -- also known as the midline or midsegment. This is a line that runs halfway between the two bases in a parallel manner. Here's the formula you'll need to calculate the median:

$m=\frac{a+b}{2}$

Very straightforward! And as you might have noticed, the median is simply the average of the two bases. But why would we ever need to know the median? Well, you can use the median to calculate the area of a trapezoid with an equally simple formula:

$\mathrm{Area}=m\times h$

That's right, the area of a trapezoid is simply the height multiplied by the median.

Getting Complicated with Trapezoids

If you encounter a trapezoid on a math test, your teacher probably wants to challenge you a little. This is why you might be given only a few values, such as the height and one of the bases -- or the area but none of the sides.

This forces you to work backward and refer back to the formulas that you've just learned. As long as you remember the formulas, you can calculate almost every aspect of a trapezoid with minimal information.

You can also use your knowledge of the Pythagorean theorem to calculate missing values. Consider the following example:

You know that the median multiplied by the height gives you the area. Using $a^2+b^2=c^2$ ,you can find out the height:

$5^2+b^2={13}^2$

$b=12\hspace{0.17em}m$

Now that we know the height, we can multiply it by the median:

$A=12\times 12$

$A=144\hspace{0.17em}{m}^2$

Topics related to the Trapezoid

Midsegment of a Trapezoid

Congruent Tangents and Circumscribed Polygons

Tessellations

Flashcards covering the Trapezoid

Advanced Geometry Flashcards

Common Core: High School - Geometry Flashcards

Practice tests covering the Trapezoid

Common Core: High School - Geometry Diagnostic Tests

Advanced Geometry Diagnostic Tests

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