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What do a Chinese takeaway box and a designer handbag have in common? Observe how they represent the same shape.
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.
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.
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 nonparallel 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 nonparallel 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 midsection) of a trapezoid is the line segment connecting the midpoints of the two nonparallel 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.
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 
Parallelogram Parallelogram, StudySmarter Originals 

Square Square, StudySmarter Originals 

Rectangle, StudySmarter Originals 

There are five types of trapezoids we should consider, namely
Scalene trapezoid
Isosceles trapezoid
Right trapezoid
Obtuse trapezoid
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 nonparallel 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 90^{o}). 
Obtuse Trapezoid  Obtuse trapezoid, StudySmarter Originals  A trapezoid with two opposite obtuse angles (more than 90^{o}). 
Acute Trapezoid  Acute trapezoid, StudySmarter Originals  A trapezoid with two adjacent acute angles (less than 90^{o}). 
A trapezoid is a twodimensional polygon that lies on a twodimensional 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.
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.
$P=13+21+19+34\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\Rightarrow P=87units$
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.
$P=5+8+x+x\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\Rightarrow 35=13+2x$
Rearranging this equation, we obtain
$2x=3513\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\Rightarrow 2x=22$
Simplifying this, we obtain
$x=\frac{22}{2}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\Rightarrow x=11units$
Thus, the value of each opposite side is 11 units.
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
$A=\frac{1}{2}(a+b)\times h$ ,
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
$A=\left(\frac{BC+AD}{2}\right)\times h$.
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
$A=\frac{1}{2}(a+b)\times h\phantom{\rule{0ex}{0ex}}\Rightarrow A=\frac{1}{2}(6+8)\times 5$
Simplifying this, we get a final answer of
$A=\frac{1}{2}\left(14\right)\times 5\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\Rightarrow A=7\times 5\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\Rightarrow A=35unit{s}^{2}$
Thus, the area of this trapezoid is 35 units^{2}.
Find the length of the shorter base of a trapezoid given that the area is 232 units^{2}, the height is 16 units and the length of the longer base is 17 units.
Solution
In this case,
A = 232 units^{2}
b = 17 units;
h = 16 units.
Substituting these values into our formula, we obtain
$A=\frac{1}{2}(a+b)\times h\phantom{\rule{0ex}{0ex}}\Rightarrow 232=\frac{1}{2}(a+17)\times 16$
Solving this, we have
$232=\frac{16(a+17)}{2}\phantom{\rule{0ex}{0ex}}\Rightarrow 232=8(a+17)$
Expanding this, we get
$232=8a+136\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\Rightarrow 8a+136=232$
Rearranging this equation and solving for a, we obtain the following final answer.
$8a=232136\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\Rightarrow 8a=96\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\Rightarrow a=\frac{96}{8}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\Rightarrow a=12units$
Hence, the length of the shorter base of this trapezoid is 12 units.
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 = 103^{o} and ∠BCD = 118^{o}. Both these angles are greater than 90^{o} 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
$P=AB+BC+CD+AD\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\Rightarrow P=14+16+18+22\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\Rightarrow P=70units$
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
$A=\frac{1}{2}(16+22)\times 11\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\Rightarrow A=\frac{1}{2}\left(38\right)\times 11\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\Rightarrow A=19\times 11\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\Rightarrow A=209unit{s}^{2}$
Thus, the area of this trapezoid is 209 units^{2}.
Bonus Question
What is the value of angle ∠ADC given that ∠ABC = 88^{o}?
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.
$\angle ABC+\angle BCD+\angle ADC+\angle BAD=360\xb0\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\Rightarrow 88\xb0+118\xb0+\angle ADC+103\xb0==360\xb0$
Rearranging this and solving for the unknown angle, we obtain
$\angle ADC=360\xb088\xb0118\xb0103\xb0\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\Rightarrow \angle ADC=51\xb0$
Thus, angle ∠ADC is 51^{o}.
A quadrilateral with one set of parallel sides.
The main characteristics of a trapezoid are:
No, not all trapezoids are parallelograms.
A right trapezoid is an example of a trapezoid.
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