Area of Trapezoid

• Introduction
• What is a Trapezoid
• Properties of a Trapezoid
• Types of Trapezoid
• Formulas of Trapezoid
• Area of Trapezoid `(A)`
• Derivation of Area of Trapezoid
• Perimeter of Trapezoid `(P)`
• The Height of the Trapezoid `(h)`
• Midsegment of Trapezoid `(m)`
• Practice Problems
• Frequently Asked Questions

Introduction

A trapezoid is a special kind of a quadrilateral. The defining feature of a trapezoid is the presence of one pair of parallel sides. These parallel sides are called the bases of the trapezoid. The other two sides of the trapezoid, which are not parallel, are known as the legs.

The distance between these parallel sides is crucial and is termed the altitude of the trapezoid. The altitude is measured as the perpendicular distance between the two parallel bases.

Other names of trapezoid are:

• Trapezium
• Trapez
• Irregular Quadrilateral

What is a Trapezoid

When we talk about what is a trapezoid, we're essentially describing a quadrilateral with a unique property: two opposite sides are parallel, but the other two opposite sides are not. This creates a distinctive shape that often appears in various mathematical problems and real-world scenarios. By breaking down a trapezoid into its defining characteristics, we can better understand its role in geometry and beyond.

Properties of a Trapezoid

`1`. Parallel Sides: A trapezoid always has two pairs of opposite sides, but only one pair is parallel. The other pair of sides is not parallel.

`2`. Bases and Legs: The parallel sides in a trapezoid are called the bases, while the non-parallel sides are referred to as the legs.

`3`. Altitude: The perpendicular distance between the parallel bases is known as the altitude of the trapezoid.

`4`. Sum of Angles: The sum of the interior angles of a trapezoid is always equal to \(360^\circ\).

`5`. Diagonals: A trapezoid can have one set of diagonals or both. The diagonals may or may not be of equal length.

`6`. Median: The median of a trapezoid is a line segment connecting the midpoints of the non-parallel sides. It is parallel to the bases and has a length equal to the average of the lengths of the bases.

Types of Trapezoids

Trapezoids can be classified into different types based on specific properties or characteristics. Here are a few types of trapezoids:

• Isosceles Trapezoid
• Right Trapezoid
• Scalene Trapezoid
• Cyclic Trapezoid

Isosceles Trapezoid

• Definition: An isosceles trapezoid is a trapezoid in which the legs are of equal length.
• Properties: In an isosceles trapezoid, the base angles (angles adjacent to the parallel sides) are also equal.

Right Trapezoid

• Definition: A right trapezoid is a trapezoid that contains at least one right angle (`90` degrees).
• Properties: A right trapezoid (also called a right-angled trapezoid) has two adjacent right angles.
   

Scalene Trapezoid

• Definition: A scalene trapezoid is a trapezoid in which none of the sides are of equal length.
• Properties: All four sides of a scalene trapezoid have different lengths.

Cyclic Trapezoid

• Definition: A cyclic trapezoid is a trapezoid whose vertices lie on a common circle.
• Properties: The opposite angles of a cyclic trapezoid are supplementary. A cyclic trapezoid is an isosceles trapezoid in which non-parallel sides are equal.

Formulas of Trapezoid

• Area of Trapezoid 
• Perimeter of Trapezoid
• Height of Trapezoid
• Midsegment of Trapezoid

In geometry, trapezoids are quadrilateral shapes with at least one pair of parallel sides. Understanding the formulas associated with trapezoids is crucial for solving various geometric problems involving these shapes.

Area of Trapezoid `(A)` 

The area of a trapezoid refers to the measure of the space enclosed within its boundaries. It is calculated as the total surface covered by the trapezoid in square units. To find the formula for trapezoidal area, we take the average of the lengths of the two parallel bases and multiply it by the height of the trapezoid.

Here, `b_1` and `b_2` are the lengths of the bases and `h` is the height of the trapezoid.

Derivation of Area of Trapezoid

We can derive the formula using a parallelogram.

Imagine combining two identical trapezoids along one of their non-parallel sides or legs. This new shape formed resembles a parallelogram, where the combined lengths of the trapezoids’ parallel sides (let's call them \( b_1 \) and \( b_2 \)) form the base, and the height remains the same as that of any of the identical trapezoids. (let's call it \( h \)).

Knowing that the area of a parallelogram is found by multiplying its base by its height, for our new shape, 

Parallelogram Area `=`  \( (b_1 + b_2) \times h \)

Since this is the area of two identical trapezoids we have to divide this by `2`, giving 

Trapezoid Area `= (b_1 + b_2)/2 × h` 

Thus, we assertively derive the formula for the area of a trapezoid: `A = (b_1 + b_2)/2 × h` 

Example: Consider a trapezoid with bases of length `8` units and `12` units and an altitude of `5` units. Calculate the area of the trapezoid.

Solution:

To find the area (\(A\)) of a trapezoid, we use the formula \(A = \frac{1}{2} \times (b_1 + b_2) \times h\), where \(b_1\) and \(b_2\) are the lengths of the bases and \(h\) is the altitude.

Substitute the given values into the formula:

\( A = \frac{1}{2} \times (8 + 12) \times 5 \)

\( A = \frac{1}{2} \times 20 \times 5 \)

\( A = 10 \times 5 \)

\( A = 50 \)

Therefore, the area of the trapezoid is `50` square units.

The Perimeter of Trapezoid `(P)`

The perimeter of a trapezoid refers to the total length of its boundary, encompassing all its sides.

This formula indicates that to find the perimeter of a trapezoid, we add the lengths of both parallel bases and the lengths of the two non-parallel sides (legs).

The perimeter `(P)` of a trapezoid is the total length of its boundary. It can be found by summing the lengths of all its sides:

\( P = a + b_1 + c + b_2 \)

Where:

• \( a \) and \( c \) are the lengths of the non-parallel sides, often referred to as the legs,
•  \( b_1 \) and \( b_2 \) are the lengths of the parallel bases.

Example: Find the perimeter of a trapezoid with the following side lengths:

• Length of one non-parallel side (leg), \( a = 6 \) units
• Length of the other non-parallel side (leg), \( c = 8 \) units
• Length of the first parallel base, \( b_1 = 10 \) units
• Length of the second parallel base, \( b_2 = 12 \) units

Solution:

To find the perimeter `(P)` of the trapezoid, we sum the lengths of all its sides.

We use the formula for perimeter:

\( P = a + b_1 + c + b_2 \)

Substituting the given values:

\( P = 6 + 10 + 8 + 12 \)

\( P = 36 \)

Therefore, the perimeter of the trapezoid is \( 36 \) units.

Height of Trapezoid `(h)`

The height of a trapezoid refers to the perpendicular distance between its two parallel bases.

To determine the height of a trapezoid, we multiply the area of the trapezoid by `2` and divide the result by the sum of the lengths of the two parallel bases.

If the area and the lengths of the bases of a trapezoid are known, the height `(h)` can be determined using the area formula rearranged as

\( h = \frac{2A}{b_1 + b_2} \)

Where:

\( A \) is the area of the trapezoid,

\( b_1 \) and \( b_2 \) are the lengths of the two parallel bases.

Example: Determine the height of a trapezoid if its area is \( 72 \) square units and the lengths of its bases are \( 6 \) units and \( 10 \) units.

Solution:

To find the height (\( h \)) of the trapezoid, we can use the formula for area of a trapezoid rearranged to solve for height.

Given:

•  Area of the trapezoid, \( A = 72 \) square units
• Length of the first base, \( b_1 = 6 \) units
• Length of the second base, \( b_2 = 10 \) units

We use the formula:

\( h = \frac{2A}{b_1 + b_2} \)

Substituting the given values:

\( h = \frac{2 \times 72}{6 + 10} \)

\( h = \frac{144}{16} \)

\( h = 9 \)

Therefore, the height of the trapezoid is \( 9 \) units.

Midsegment of Trapezoid `(m)`

The midsegment of a trapezoid (also called the median of a trapezoid) is a line segment that connects the midpoints of the non-parallel sides (legs). It divides the trapezoid into two smaller trapezoids, each having the same altitude (height), but not necessarily the same area. To find the length of the midsegment of a trapezoid, we add the lengths of both parallel bases and then divide the sum by `2`. Its length `(m)` can be calculated using the formula:

\( m = \frac{b_1 + b_2}{2} \)

Here:

\( b_1 \) and \( b_2 \) are the lengths of the parallel bases.

Example `1`: Calculate the length of the midsegment of a trapezoid if the lengths of its bases are \( 12 \) units and \( 8 \) units, respectively.

Solution:

To find the length of the midsegment (\( m \)) of the trapezoid, we use the formula for the midsegment of a trapezoid.

Given:

• Length of the first base, \( b_1 = 12 \) units
• Length of the second base, \( b_2 = 8 \) units

We use the formula:

\( m = \frac{b_1 + b_2}{2} \)

Substituting the given values:

\( m = \frac{12 + 8}{2} \)

\( m = \frac{20}{2} \)

\( m = 10 \)

Therefore, the length of the midsegment of the trapezoid is \( 10 \) units.

Example `2`: A trapezoid has a perimeter of \( 36 \) units. The lengths of its non-parallel sides (legs) are \( 7 \) units and \( 9 \) units, and its height is \( 4 \) units. What is the area of the trapezoid?

Solution:

To find the area (\( A \)) of the trapezoid, we need to use the formula for the area of a trapezoid:

\( A = \frac{1}{2} \times (b_1 + b_2) \times h \)

Given:

• Perimeter of the trapezoid, \( P = 36 \) units
• Length of one leg, \( a = 7 \) units
• Length of the other leg, \( c = 9 \) units
• Height of the trapezoid, \( h = 4 \) units

First, let's find the lengths of the parallel bases (\( b_1 \) and \( b_2 \)) using the perimeter:

\( P = a + b_1 + c + b_2 \)

\( 36 = 7 + b_1 + 9 + b_2 \)

\( 36 = 16 + b_1 + b_2 \)

\( b_1 + b_2 = 36 - 16 \)

\( b_1 + b_2 = 20 \)

Substituting the value of \( b_1 + b_2\) and `h`  into the formula for the area:

\( A = \frac{1}{2} \times 20 \times 4 \)

\( A = 10 \times 4 \)

\( A = 40 \)

Therefore, the area of the trapezoid is \( 40 \) square units.

Practice Problem

Q`1`. An isosceles trapezoid has bases of length `10` cm and `14` cm. If the altitude is `6` cm, use trapezoidal area formula to find the area of the trapezoid.

a) `19` square centimeters

b) `72` square centimeters

c) `52` square centimeters

d) `42` square centimeters

Answer: b

Q`2`. Calculate the perimeter of a trapezoid if the lengths of its non-parallel sides (legs) are \( 6 \) units and \( 8 \) units, and the lengths of its parallel bases are \( 10 \) units and \( 14 \) units.

a) `24` units

b) `52` units

c) `31` units

d) `38` units

Answer: d

Q`3`. An irregular trapezoid has bases of length `15` cm and `9` cm, with an altitude of `8` cm. Calculate the area of the trapezoid.

a) `96` square cm  

b) `120` square cm  

c) `108` square cm  

d) `80` square cm  

Answer: a

Q`4`. Find the perimeter of a trapezoid if the lengths of its non-parallel sides (legs) are \( 5 \) units and \( 7 \) units, and the lengths of its parallel bases are \( 9 \) units and \( 12 \) units.

a) \( 33 \) units  

b) \( 35 \) units  

c) \( 38 \) units  

d) \( 41 \) units  

Answer: a

Frequently Asked Questions

Q`1`. Are all trapezoids quadrilaterals?

Answer: Yes, all trapezoids are quadrilaterals. A trapezoid is defined as a four-sided polygon with one pair of parallel sides. Therefore, it falls under the category of quadrilaterals, which are polygons with four sides.

Q`2`. What is the difference between a trapezoid and a trapezium?

Answer: In some regions, the terms "trapezoid" and "trapezium" are used interchangeably, but in others, they have distinct meanings. In the United States, a trapezoid has one pair of parallel sides, while a trapezium is a quadrilateral with no parallel sides. In other English-speaking countries, a trapezoid is any quadrilateral with at least one pair of parallel sides, and a trapezium is a specific type of trapezoid with no parallel sides.

Q`3`. Can a trapezoid have `4` right angles?

Answer: No, a trapezoid cannot have all four angles as right angles. A trapezoid is a quadrilateral with at least one pair of parallel sides. The parallel sides are called bases, and the non-parallel sides are called legs. A trapezoid may have both pairs of opposite sides unequal in length. If all four angles were right angles, the sides would be parallel, making it a parallelogram, not a trapezoid.

Q`4`. Are all rhombuses also trapezoids?

Answer: Yes, all rhombuses are trapezoids. A rhombus is a type of quadrilateral with all sides of equal length. Since a trapezoid is defined by having at least one pair of parallel sides, and a rhombus has opposite sides that are parallel, a rhombus can be considered a special type of trapezoid.

Q`5`. Is a trapezoid a parallelogram?

Answer: Certainly! Here are the exclusive and inclusive definitions of a trapezoid as a parallelogram:

Exclusive Definition:

“A trapezoid is a quadrilateral with exactly one pair of parallel sides.”

(This definition specifically states that a trapezoid must have only one pair of parallel sides, excluding parallelograms, rectangles, and squares, which all have two pairs of parallel sides.)

Inclusive Definition:

“A trapezoid is a quadrilateral with at least one pair of parallel sides.”

(This definition broadens the concept of a trapezoid to include shapes with more than one pair of parallel sides, encompassing parallelograms as well. It acknowledges that all parallelograms are trapezoids, but not all trapezoids are parallelograms.)