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• Trapezoid – Definition, Types, Properties, Examples, FAQs

Trapezoid – Definition, Types, Properties, Examples, FAQs

Euclid, an ancient Greek mathematician, was the one who named the five quadrilaterals. They  were square, rectangle, rhombus, and rhomboid with four sets of parallel sides, while the last one was a trapezia which had only two sets of parallel sides. Later, Proclus introduced two types of trapezia. One was the trapezium and had only two pairs of parallel lines, while another one was the trapezoid which had one or no pair of opposite parallel lines. 

Till date, there is still confusion whether trapezoid and trapezium are the same thing. Basically, they have the same shape with different structures because a trapezoid is trapezium-like with no opposite parallel sides.

What Is a Trapezoid?

A trapezoid is a quadrilateral with one pair of parallel sides. It is a 2D flat closed shape with four sides. It has one set of opposite parallel sides known as the base and another set of non-parallel sides known as the legs or laterals of a trapezoid. The perpendicular distance between the parallel sides is known as altitude. The sum of all the interior angles adds up to 360 degrees. 

There are still disagreements on the definition of a trapezoid. Some schools say that a trapezoid can have only one or no pair of parallel sides, while others say that it can have more than one pair of parallel sides. According to the second description, a parallelogram qualifies to be a trapezoid, but the first description disqualifies this statement. To come to an agreement, we can say that the exclusive definition of a trapezoid states that it has no opposite parallel sides, while the inclusive one states that it has at least one pair of opposite parallel sides.

Types of Trapezoids

Trapezoids can be classified into three groups:

1. Right trapezoid
2. Isosceles trapezoid
3. Scalene trapezoid

Right trapezoid

A right trapezoid is a trapezoid with one of its legs perpendicular to both of the bases. It is also called a right-angled trapezoid since it has a right angle of 90 degrees. Since there is a right angle formed at one of the bases, automatically, another right angle is formed at the other base.

Isosceles trapezoid

An isosceles trapezoid is a trapezoid with its top and bottom as parallel lines, while the remaining pair of lines are non-parallel and have equal length. The diagonals and the base angles of an isosceles trapezoid are equal. Similar to the properties of other quadrilaterals, the interior angles of an isosceles trapezoid add up to 360 degrees. 

Scalene trapezoid

A scalene trapezoid is also called a non-isosceles trapezoid. It has sides that are not congruent and have different lengths. Due to this feature, a scalene trapezoid forms four angles of different measures at the vertices. A scalene trapezoid exhibits all properties of a trapezoid since it has two parallel sides and two non-parallel sides, even though they have different lengths.

Properties of a Trapezoid

Though trapezoids are quadrilaterals, they have properties that make them stand out from other quadrilaterals. Below are some trapezoid properties: 

• The bases (the top and bottom line) are parallel to each other. Parallel means they follow the same path at different points but can never meet.
• The opposite sides of an isosceles are of the same length.
• Trapezoids have a median which is parallel to both bases.
• A trapezoid is said to be a parallelogram if both opposite sides are parallel to each other.
• In a case where a trapezoid has all sides of equal length and at right angles with each other and both pairs of sides are parallel, then it can be called a square.
• A trapezoid can also be a rectangle if opposite sides are parallels of equal length and meet at a right angle.
• The sum of angles on the same side add up to 180 degrees, and the sum of all angles  add to 360 degrees.

Median of a Trapezoid

The median of a trapezoid is parallel to both its bases and is equal to half the sum of both the bases.

Area of a Trapezoid

The area of a trapezoid is the total area covered within the trapezoid borders.

Area= ½ (A+B)h, where A and B are the bases, while h is the height.

This trapezoid area formula is derived from the area of a parallelogram. Suppose we rotate an existing trapezoid shape and adjoin the existing trapezoid and the rotated trapezoid. The figure formed will be a parallelogram as shown below

Area of a parallelogram = (A+B)h, hence the area of a trapezoid is half the area of the parallelogram because a trapezoid is half of a parallelogram.

The unit area of a trapezium is defined as =ft2 cm2  m2

Perimeter of a Trapezoid

Perimeter is the total distance covered round the trapezoid. Hence, the perimeter of a trapezoid can be calculated by adding the lengths of all four sides. 

This formula only applies when the lengths of all the sides are given. When some sides are missing, we use the Pythagorean theorem. The formula uses the given relevant information to find the lengths of the missing sides. 

The Pythagorean theorem states that C2  = a2+ b2, where C is the hypotenuse, a is the height, and b is the base

As seen from the figure above, we can use the Pythagorean theorem to find the length of side C

Since we now have the length of side C, the perimeter of the trapezoid can be calculated            

P=16+5+10+5=36cm

Solved Examples on Trapezoids 

Example 1

Calculate the area of the given trapezoid

Area= ½ (a+b)h, where a and b are the bases, h is the height

Area=½ (20+32)10    

Area=½ 5210  

Area= 260m2

Example 2

Calculate the perimeter of the following trapezoid:

P =a+b+c+d

P = 15cm + 35cm + 14cm + 20c = 84cm

Frequently Asked Questions ( FAQ)

Are all trapezoids quadrilaterals?

According to the trapezoid definition, they are quadrilaterals with one pair of opposite sides that are parallel. Trapezoids fit to be quadrilaterals since they have the following quadrilateral properties:

• They have four vertices.
• The sum of interior angles adds up to 360 degrees.
• They have four sides and two diagonals.

What is the difference between trapezoid and trapezium?

The major difference between a trapezoid and a trapezium is that a trapezoid can have no pair of parallel opposite sides, while a trapezium has at least one pair of parallel opposite sides. This means that if a trapezoid has one pair of parallel opposite sides, it can be called a trapezium.

Can a trapezoid have four right angles?

As mentioned in the properties of trapezoids, a square is a special trapezoid formed when all opposite sides are parallel and are of equal length. This means all sides meet at right angles of 90 degrees. Hence, a trapezoid can have four right angles as long as it is a square. 

Are all rhombuses also trapezoids?

Not all rhombuses are trapezoids because a trapezoid has only one pair of parallel opposite sides and rhombuses have two pairs of opposite parallel sides that are equal in length. 

Is trapezoid a parallelogram?

A trapezoid cannot be a parallelogram because all pairs of opposite sides are parallel to each other and a trapezoid has only one pair of parallel opposite sides. In other words, a parallelogram can be said to be a trapezoid and not the other way round.

Conclusion

Trapezoids can be noticed in the real world in the forms of roofs, windows, table tops, houses, and pencil boxes. The most common objects with the trapezoid shape are flowerpots, the shade cap of a lamp, buckets, handbags, popcorn tubs, and guitars. Items like guitars have an irregular trapezoidal shape, while buckets and popcorn tubs are regular shapes. Most real-life trapezoidal objects are three dimension geometric shapes. Educators are advised to use the inclusive definition of trapezoids that includes parallelograms as a trapezoid.

A learner with the knowledge of a trapezoid will easily learn the topic of parallelograms since the properties are also true for parallelograms. Just like with other shapes, learning trapezoids helps enhance the foundational cognitive ability and visualization skills in kids who learn math.