# Trapezoids

What do a Chinese takeaway box and a designer handbag have in common? Observe how they represent the same shape.

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Parallelogram shape of a handbag and takeaway box, StudySmarter Originals

Now, notice how both the bases of the handbag and takeaway box are parallel to their tops. Since this shape has four sides, it is classified as a type of quadrilateral. However, it is neither a square, a rectangle nor a parallelogram. These shapes have two pairs of parallel sides while the shape described by this handbag and takeaway box has only one pair. Have you got any guesses as to what this shape might be? Let me give you a hint: it's called a trapezoid.

This article will explore the definition of a trapezoid along with its characteristics and types. We shall also look into the formulas used to find the perimeter and area of a trapezoid.

## 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.

A trapezoid is a quadrilateral with one set of parallel sides.

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;

By the definition of a trapezoid.

• 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 median (or mid-section) of a trapezoid is the line segment connecting the midpoints of the two non-parallel sides of a trapezoid.

• 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.

 Type of Quadrilateral Description ParallelogramParallelogram, StudySmarter Originals A trapezoid where both pairs of opposite sides are parallel to each other SquareSquare, StudySmarter Originals A trapezoid where both pairs of opposite sides are parallel to each other All four sides are of equal length and at right angles to each other RectangleRectangle, StudySmarter Originals A trapezoid where both pairs of opposite sides are parallel to each otherThe opposite sides are of equal length and at right angles to each other

## 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.

 Type of Trapezoid Pictorial Representation Description Scalene Trapezoid Scalene trapezoid, StudySmarter Originals A trapezoid with no sides or angles of equal measure. Isosceles Trapezoid Isosceles trapezoid, StudySmarter Originals A trapezoid with opposite sides of the same length. Usually, represented by the non-parallel sides (or legs) of a trapezoid. The angles of the parallel sides (or bases) are equal to each other. Right Trapezoid Right trapezoid, StudySmarter Originals A trapezoid with two adjacent right angles (equal to 90o). Obtuse Trapezoid Obtuse trapezoid, StudySmarter Originals A trapezoid with two opposite obtuse angles (more than 90o). Acute Trapezoid Acute trapezoid, StudySmarter Originals A trapezoid with two adjacent acute angles (less than 90o).

## 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.

Given the trapezoid below, find its perimeter.

Example 1, StudySmarter Originals

Solution

To find the perimeter of this trapezoid, we shall simply add the measures of all four sides together.

$P=13+21+19+34\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}⇒P=87units$

Thus, the perimeter of this trapezoid is 87 units.

An isosceles trapezoid has a perimeter of 35 units. What is the length of each (equal) opposite side given that the bases are 5 units and 8 units, respectively?

Solution

Here, we are given the perimeter of a trapezoid and the lengths of the bases. We are also told that this trapezoid is an isosceles trapezoid, meaning that there is a pair of equal opposite sides. We shall name these two identical sides by x.

Example 2, StudySmarter Originals

Since the perimeter is the sum of all four sides of this trapezoid, we can write this as the equation below.

$P=5+8+x+x\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}⇒35=13+2x$

Rearranging this equation, we obtain

$2x=35-13\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}⇒2x=22$

Simplifying this, we obtain

$x=\frac{22}{2}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}⇒x=11units$

Thus, the value of each opposite side is 11 units.

## 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.

The height, h of a trapezoid is at a perpendicular distance between bases, a and b. It is also referred to as the altitude of a trapezoid.

Thus, the area of a trapezoid is

$A=\frac{1}{2}\left(a+b\right)×h$ ,

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

$A=\left(\frac{BC+AD}{2}\right)×h$.

### Examples Using the Area of a Trapezoid Formula

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

Identify the area of the following trapezoid.

Example 3, StudySmarter Originals

Solution

Here,

a = 6 units;

b = 8 units;

h = 5 units.

Don't get yourselves confused by the other two sides given! They are not parallel to each other so we cannot use their measures in our formula.

Now, using the area of a trapezoid formula, we obtain

$A=\frac{1}{2}\left(a+b\right)×h\phantom{\rule{0ex}{0ex}}⇒A=\frac{1}{2}\left(6+8\right)×5$

Simplifying this, we get a final answer of

$A=\frac{1}{2}\left(14\right)×5\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}⇒A=7×5\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}⇒A=35unit{s}^{2}$

Thus, the area of this trapezoid is 35 units2.

Find the length of the shorter base of a trapezoid given that the area is 232 units2, the height is 16 units and the length of the longer base is 17 units.

Solution

In this case,

A = 232 units2

b = 17 units;

h = 16 units.

Substituting these values into our formula, we obtain

$A=\frac{1}{2}\left(a+b\right)×h\phantom{\rule{0ex}{0ex}}⇒232=\frac{1}{2}\left(a+17\right)×16$

Solving this, we have

$232=\frac{16\left(a+17\right)}{2}\phantom{\rule{0ex}{0ex}}⇒232=8\left(a+17\right)$

Expanding this, we get

$232=8a+136\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}⇒8a+136=232$

Rearranging this equation and solving for a, we obtain the following final answer.

$8a=232-136\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}⇒8a=96\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}⇒a=\frac{96}{8}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}⇒a=12units$

Hence, the length of the shorter base of this trapezoid is 12 units.

## Example Involving Trapezoids

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

Given the trapezoid ABCD below, determine its type, perimeter and area.

Example 4, StudySmarter Originals

Solution

Type

Let us first deduce what type of trapezoid this is. Looking at the diagram above, observe that ∠BAD = 103o and ∠BCD = 118o. Both these angles are greater than 90o and are located opposite each other. Thus, we have an obtuse trapezoid.

Perimeter

Next, we shall find the perimeter of this trapezoid. Adding all four sides of this trapezoid, we obtain

$P=AB+BC+CD+AD\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}⇒P=14+16+18+22\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}⇒P=70units$

Thus, the perimeter of this trapezoid is 70 units.

Area

Here, BC (shorter base) is parallel to AD (longer base). The height is perpendicular to both these bases. Thus,

a = 16 units;

b = 22 units;

h = 11 units.

Applying the formula of the area of a trapezoid, we obtain

$A=\frac{1}{2}\left(16+22\right)×11\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}⇒A=\frac{1}{2}\left(38\right)×11\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}⇒A=19×11\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}⇒A=209unit{s}^{2}$

Thus, the area of this trapezoid is 209 units2.

Bonus Question

What is the value of angle ∠ADC given that ∠ABC = 88o?

By the property of trapezoids, the sum of its interior angles adds up to 360°. Since we have the measures of three angles, we can find the value of the missing angle below.

$\angle ABC+\angle BCD+\angle ADC+\angle BAD=360°\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}⇒88°+118°+\angle ADC+103°==360°$

Rearranging this and solving for the unknown angle, we obtain

$\angle ADC=360°-88°-118°-103°\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}⇒\angle ADC=51°$

## Trapezoids - Key takeaways

• A trapezoid is a quadrilateral with one set of parallel sides.
• There are 5 types of trapezoids: scalene, isosceles, right, obtuse and acute.
• The perimeter of a trapezoid is given by P = a + b + c + d.
• The area of a trapezoid is given by $A=\frac{1}{2}\left(a+b\right)×h$.
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What is a trapezoid?

A quadrilateral with one set of parallel sides.

What are the characteristics of trapezoids?

The main characteristics of a trapezoid are:

• it has a pair of parallel sides;
• it has a pair of adjacent angles formed between one parallel side and one non-parallel side;
• its diagonals bisect each other;
• its median is parallel to the parallel sides.

Are all trapezoids parallelograms?

No, not all trapezoids are parallelograms.

What is an example of a trapezoid?

A right trapezoid is an example of a trapezoid.

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