Trapezoids

What is a Trapezoid?

As mentioned before, a trapezoid falls under the category of a quadrilateral as it contains four sides. This special type of quadrilateral actually has two names: a trapezoid and a trapezium. The name varies from where you are in the world. Here in the United States, it is typically called a trapezium. However, in the United Kingdom, it is usually called a trapezium. How interesting is that? With that in mind, let us begin our discussion with the definition of a trapezoid.

Below is a graphical representation of a trapezoid. We shall call this trapezoid ABCD.

Illustration of a trapezoid, StudySmarter Originals

We shall now move on to listing the properties of a trapezoid. By doing so, we can observe how different they are compared to a regular quadrilateral.

Characteristics of a Trapezoid

Let us now refer back to our trapezoid ABCD above. There are several notable characteristics of trapezoids we should familiarize ourselves with. These are listed below.

• A trapezoid has a pair of parallel sides and a pair of non-parallel sides;

• Usually, the bases (the top and bottom) of ABCD are parallel to each other. This can be written as AD // BC;

• A pair of adjacent angles formed between one parallel side and one non-parallel side of a trapezoid add up to 180°. Here, ∠ABC + ∠BAD = 180° and ∠BCD + ∠ADC = 180°;

• The sum of the interior angles of a trapezoid is 360°;

• The diagonals of a trapezoid bisect each other;

• The median (midline or midsegment) of a trapezoid is parallel to both bases. This is shown by the pink line below;

Median of a trapezoid, StudySmarter Originals

• The length of the median is the average of both bases. Say a = AD and b = BC, then $m=\frac{a+b}{2}$ , where m is the median.

Forming Other Quadrilaterals from Trapezoids

There are three types of quadrilaterals that can stem from a trapezoid, namely a parallelogram, a square and a rectangle. These instances are described in the table below.

Types of Trapezoids

There are five types of trapezoids we should consider, namely

1. Scalene trapezoid

2. Isosceles trapezoid

3. Right trapezoid

4. Obtuse trapezoid

5. Acute trapezoid

The table below describes each of these trapezoids in turn along with their pictorial representation and distinct traits.

The Perimeter of a Trapezoid

A trapezoid is a two-dimensional polygon that lies on a two-dimensional plane. The perimeter of a trapezoid is described as the total length of its boundary. In other words, it is the sum of all its sides. Say we have a trapezoid ABCD with sides a, b, c, and d.

The perimeter of a trapezoid, StudySmarter Originals

Then the perimeter of a trapezoid formula is

P = a + b + c + d,

where P is the perimeter, a = AB, b = BC, c = CD and d = AD. This can also be written as

P = AB + BC + CD + AD.

Examples Using the Perimeter of a Trapezoid Formula

Let us now look at some worked examples involving the formula for finding the perimeter of a trapezoid.

The Area of a Trapezoid

The area of a trapezoid is defined by the space enclosed within its boundary. It is found by calculating the average length between two given parallel sides and multiplying this product with the height of the trapezoid. Observe the illustration of trapezoid ABCD below.

Area of a trapezoid, StudySmarter Originals

Here, the bases are a = BC and b = AD. The height is denoted by the letter h.

Thus, the area of a trapezoid is

$A=\frac{1}{2}(a+b)\times h$ ,

where A = area, a = length of the shorter base, b = length of the longer base and h = height. Similarly, we can express this formula as

$A=\left(\frac{BC+AD}{2}\right)\times h$.

Examples Using the Area of a Trapezoid Formula

Let us now look at some worked examples applying the area of a trapezoid formula.

Example Involving Trapezoids

We shall end this topic with an example that encompasses everything we have learnt throughout this discussion.