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# Trapezoid

• properly define a trapezoid;
• know the exclusive and inclusive definition of a trapezoid;
• differentiate a trapezoid from a trapezium;
• learn the parts of a trapezoid;
• state all the properties of a trapezoid;
• determine the different types of trapezoid;
• calculate the perimeter of a trapezoid; and
• solve for the area of a trapezoid.

## What is a trapezoid?

In Euclidean geometry, a convex quadrilateral with at least one pair of parallel sides is called a trapezium outside North America but a trapezoid in American and Canadian English.

A trapezoid is a closed two-dimensional figure with four sides. Two parallel sides of a trapezoid are referred to as the bases of the trapezoid. The non-parallel sides of a trapezoid are referred to as the legs or lateral sides. The altitude is the shortest distance between two parallel sides.

The figure below shows the parts and a sample of how a trapezoid looks like.

### Exclusive definition

There is some debate over whether parallelograms, which have two parallel pairs of sides, should be considered trapezoids. According to some, a trapezoid is defined as a quadrilateral with exactly one pair of parallel sides, excluding parallelograms.

### Inclusive definition

Others define a trapezoid as a quadrilateral with at least one pair of parallel sides, classifying the parallelogram as a subtype of a trapezoid. This definition corresponds to its application in higher mathematics, such as calculus.

All parallelograms (including rhombuses, rectangles, and squares) are trapezoids by the inclusive definition. Rectangles have mirror symmetry on their mid-edges; rhombuses have mirror symmetry on their vertices, and squares have both mid-edges and vertices.

## Is it a trapezoid or trapezium?

Euclid, an ancient Greek mathematician, defined five types of quadrilaterals, four of which had two sets of parallel sides: square, rectangle, rhombus, and rhomboid. And one of which did not have two sets of parallel sides – a τραπέζια. Trapezia means “a table,” from (tetrás), “four,” and (péza), “a foot; end, border, edge”.

Proclus (412–485 AD) introduced two types of trapezia in his commentary on the first book of Euclid’s Elements:

• one pair of parallel sides – a trapezium (τραπέζιοv), which can be classified as isosceles (equal legs) or scalene (unequal legs) trapeziums; and
• no parallel sides – a trapezoid (τραπεζoειδή, trapezoeidé, literally trapezium-like (εἶδος means “resembles”) in the same way that cuboid and rhomboid both mean cube-like.

All European languages, including English, followed Proclus’s structure until the late 18th century when an influential mathematical dictionary published in 1795 by Charles Hutton implicitly supported a transposition of the terms. This error was corrected in British English around 1875, but remained in American English until the early twentieth century.

Hutton’s exact definition of trapezium and trapezoid in 1795 in a book, A Philosophical and Mathematical Dictionary: Containing an Explanation of Terms, is defined below:

• Trapezium, in Geometry, a plane figure of four straight sides, of which opposites are not parallel. When this figure has two of its sides parallel to each other, it is sometimes called a trapezoid.
• Trapezoid, sometimes denotes a trapezium that has two of its sides parallel to each other; and sometimes an irregular figure, having four sides not parallel to each other.

## What are the parts of a trapezoid?

The parts of a trapezoid are its base, leg or lateral side, height, midsegment, and base angles.

• The bases of a trapezoid are the pair of parallel lines.
• The legs or lateral sides are the pair of non-parallel sides of the trapezoid.
• The height or altitude is the distance between the two bases.
• The base angle is the interior angle by the same side of the trapezoid.
• The midsegment is the segment that connects the midpoints of the legs that is always parallel with the base.

## What are the properties of a trapezoid?

The following properties imply and distinguish a convex trapezoid from other quadrilaterals:

• It has two adjacent angles that are supplementary, which means the sum of two angles is 180°.
• The angle formed by the side and the diagonal is equal to the angle formed by the opposite side and the same diagonal.
• The diagonals cut each other in the same ratio.
• The diagonals cut the quadrilateral into four triangles, each of which is similar to one of its opposite pairs.
• The diagonals cut the quadrilateral into four triangles, each with an equal area.
• The product of the area of two triangles formed by one diagonal is equal to the product of the area of two triangles formed by the other diagonal.
• The median of a trapezoid is parallel to both bases.
• If both pairs of opposite sides of a trapezoid are parallel, it is referred to as a parallelogram.
• When both pairs of opposite sides are parallel, all sides are equal in length, and all sides are perpendicular to one another, then a trapezoid is called a square.
• If both pairs of opposite sides are parallel, equal in length, and perpendicular to one another, a trapezoid is considered a rectangle.

## What are the types of trapezoid?

There are three types of trapezoid namely; isosceles trapezoid, scalene trapezoid, and right trapezoid.

### Isosceles trapezoid

When the legs or non-parallel sides of a trapezoid are equal in length, the trapezoid is said to be isosceles. The parallel sides (base) of the isosceles trapezoid have equal angles. A trapezoid that is isosceles has a line of symmetry and both diagonals are equal in length.

In the isosceles trapezoid ABCD, AB and CD are referred to as the bases of the isosceles trapezoid. AC and BD are referred to as the trapezoid’s legs because they are not parallel to one another.

#### Properties of an Isosceles Trapezoid

The properties of an isosceles trapezoid are as follows:

• It has an axis of symmetry. It does not have a rotational symmetry and only has one line of symmetry connecting the parallel sides’ midpoints.
• One pair of parallel sides is referred to as the base sides. In trapezoid ABCD, the pair of parallel sides are AB and CD.
• Apart from the base, the remaining sides are non-parallel and equal in length. The legs or lateral sides of trapezoid ABCD are AC and CD.
• Diagonals are equal in length. The diagonals of trapezoid ABCD are AD and CB.
• The angles at the base are identical. In trapezoid ABCD, the base angles are ∠A and ∠B are equal.
• The sum of opposite angles is supplementary or equal to 180°. Thus, in trapezoid ABCD, ∠A+ ∠D=180° and ∠B+ ∠C=180°.
• The line segment connecting the parallel sides’ midpoints is perpendicular to the bases. Thus in trapezoid ABCD, if a line segment MP connects sides AB and CD, then, the angle formed P is equal to 90°.

### Scalene trapezoid

When the trapezoid’s sides and angles are not equal, it is called a scalene trapezoid.

In the scalene trapezoid EFGH, each of the four sides, EF, FH, GH, and EG, are of different lengths. The bases, EF and GH, are parallel but differ in length.

### Right trapezoid

A right trapezoid (alternatively referred to as a right-angled trapezoid) is a polygon with two adjacent right angles. The trapezoidal rule is used to determine the area under a curve.

In the given right trapezoid, or right-angled trapezoid LMNO, the right angles are at L and N. The pair of parallel sides are DC and AB.

## How find the perimeter of a trapezoid?

The perimeter of a trapezoid is equal to the sum of the lengths of each of its sides. The perimeter of the trapezoid is specified in units, which can be in., ft., m., or cm., for example.

To find the perimeter of a trapezoid:

Step 1: Determine the dimensions of all the trapezoid’s sides.
Step 2: Add the measures of all the sides.
Step 3: Once you find the perimeter of a trapezoid, write the appropriate measurement of units.

Thus, the formula of getting the perimeter of a trapezoid is:

P = a + b + c + d

where:
P is the perimeter; and
a, b, c, d are the sides of the trapezoid

Consider a trapezoid JKLM,

To calculate the perimeter of trapezoid JKLM with sides measuring 5 in., 11 in., 6 in., and 6 in., we simply add all the side measures. Thus,

PJKLM=a+b+c+d
PJKLM=5+11+6+6
PJKLM=28

Therefore, the perimeter of trapezoid JKLM is 28 inches.

Example #1

Calculate the perimeter of a trapezoid with parallel sides of measures 13 cm and 17 cm, and non-parallel sides measures of 10 cm and 11cm.

Solution

Given that:
bases of a trapezoid is equal to 13 cm and 17 cm, and
legs with measures 10 cm and 11 cm;

Thus, the perimeter of the trapezoid is defined as:

P = sum of all the sides of a trapezoid
P = 13 + 17 + 10 + 11
P = 51

Therefore, the trapezoid has a perimeter of 51 cm.

Example #2

Calculate the perimeter of a trapezoidal backyard with sides of 30 m., 12 m., 11 m., and 11 m.

Solution

Given all sides of a trapezoid with measures 30 m, 12 m., 11 m., and 11 m., then, the perimeter of a trapezoid is:

P = sum of all the sides of a trapezoid
P = 30 + 12 + 11 + 11
P = 64

Therefore, the perimeter of the trapezoidal backyard is 64 meters.

Example #3

Determine the perimeter of the trapezoid shown in the figure below.

Solution

As shown in the figure, trapezoid LMNP has side measures of 21 mm, 35 mm, 13 mm, and 10 mm. Thus, the perimeter of the trapezoid is:

P = sum of all the sides of a trapezoid
P = 21 + 35 + 13 + 10
P = 79

Therefore, the perimeter of trapezoid LMNP is 79 mm.

## How find the area of a trapezoid?

The area of a trapezoid is defined as the number of unit squares that it can contain and is expressed in square units (like cm2, m2, in2, etc).

The area of a trapezoid is determined by multiplying the average of its parallel sides by its height. Thus, the formula for solving the area of a trapezoid is given by:

A=$\frac{1}{2}$(b1+b2)(h)

where,
A is the area of the trapezoid;
b1, b2 are the bases of a trapezoid; and
h is the height of the trapezoid

Say, for example, the trapezoid ABCD,

If we are to find the area of trapezoid ABCD with bases measuring 12 cm and 8 cm, and a height of 6 cm. Thus, the area will be:

AABCD=$\frac{1}{2}$(b1+b2)(h)
AABCD=$\frac{1}{2}$(12+8)(6)
AABCD=$\frac{1}{2}$(20)(6)
AABCD=(10)(6)
AABCD =60

Therefore, the area of a trapezoid ABCD is 60 cm2.

Example #1

Find the area of a trapezoid with bases of 21 cm and 15 cm, and a height of 8 cm.

Solution

Using the formula of getting the area of a trapezoid,

A=$\frac{1}{2}$(b1+b2)(h)
A=$\frac{1}{2}$(21+15)(8)
A=$\frac{1}{2}$(36)(8)
A=(18)(8)
A=144

Therefore, the area of trapezoid is 144 cm2.

Example #2

What is the area of the trapezoid with base measures of 8 inch and 12 inch, and a height of 5 inch?

Solution

To get the area of the trapezoid, use the formula A=$\frac{1}{2}$(b1+b2)(h)). Thus,
A=$\frac{1}{2}$(8+12)(5)
A=$\frac{1}{2}$(20)(5)
A=(10)(5)
A=50

Thus, the area of the trapezoid is 50 in2.

Example #3

One of the bases of a trapezoid measures 9 meters. If the height measures 7m and the area is 77 m2, then what is the measure of the other base?

Solution

Step 1: The given information in the problem is the measure of one base, the measure of the height, and the area of the trapezoid. Hence, we need to find the measure of the other base. Thus, derive a formula using the area of the trapezoid.

Step 2: Calculate the measure of the base using the derived formula and substitute all the given information. Thus,

b2=$\frac{2A}{h}$-b1
b2=$\frac{2(77)}{7}$-9
b2=$\frac{154}{7}$-9
b2=22-9
b2=13

Therefore, the measure of the other base is 13 meters.

Example #4

If the area of a trapezoid is 160 square feet, and the measure of its parallel sides are 21 feet and 19 feet, what is the distance between the two parallel sides?

Solution:

Step 1: By definition, the distance between the two parallel sides is the height. Thus, to find the height of the trapezoid, simply derive it using the formula in finding the area of the trapezoid. Thus,

Step 2: Substitute all the given information to the formula, h=$\frac{2A}{b_1+b_2}$ . Thus,

h=$\frac{2(160)}{(21+19)}$
h=$\frac{320}{40}$
h=8

Therefore, the height of the given trapezoid is 8 square feet.

## What is a midsegment?

The midsegment (alternatively referred to as the median or midline) of a trapezoid is the segment that connects the legs’ midpoints. It is parallel to the bases of the trapezoid. The length of a midsegment is equal to the average of the bases of a trapezoid. Thus, mathematically, we can represent the midsegment as:

m= $\frac{1}{2}$(b1+b2)

where,
m is the midsegment of a trapezoid; and
b1, b2 are the bases of a trapezoid.

Say, for example, we have the lengths of the base as 40 meters and 46 meters, to find the midsegment of the trapezoid, use the formula m= $\frac{1}{2}$(b1+b2). Thus,
m= $\frac{1}{2}$(40+46)
m= $\frac{1}{2}$(86)
m=43

Therefore, the midsegment of the trapezoid is 43 meters.

## What is the significance of trapezoids?

The concepts of trapezoids can be applied in different fields aside from geometry such as architecture, biology, and computer engineering.

In geometry, the crossed ladders problem determines the distance between parallel sides of a right trapezoid given the diagonal lengths and the distance between the perpendicular leg and the diagonal intersection.

Meanwhile, in architecture, the term trapezoid refers to symmetrical doors, windows, and structures that are larger at the base and narrower at the top, in the Egyptian style. These are typically isosceles trapezoids if they have straight sides and sharp angular corners. This was the Inca’s typical design for doors and windows.

Additionally, in morphology, taxonomy, and other descriptive disciplines often use terminology like trapezoidal or trapeziform to describe certain organs or structures.

Finally, trapezoids are often used in computer engineering, particularly digital logic and computer architecture, to represent multiplexors. Multiplexors are logic components that, in response to a select signal, choose amongst several elements and generate a single output. Trapezoids are often used in designs without explicitly mentioning that they are multiplexors, since they are universally equivalent.