# Trapezium

A trapezium is a type of quadrilateral, which has only two parallel sides and the other two sides are non-parallel. In Euclidean Geometry, a quadrilateral is defined as a polygon with four sides and four vertices. Quadrilaterals are either simple or complex. The simple quadrilaterals are not self-intersecting and it is categorised as convex or concave quadrilaterals. The interior angles of the quadrilateral are equal to 360 degrees. Among these, the different types of quadrilaterals are:

• Square
• Trapezium
• Rectangle
• Parallelogram
• Rhombus

## What is Trapezium?

The trapezium is a quadrilateral with two parallel sides. The parallel sides of a trapezium are called bases and the non-parallel sides of a trapezium are called legs. It is also called as a trapezoid. Sometimes the parallelogram is also called as a trapezoid with two parallel sides.

From the above figure, we can see, sides AB and CD are parallel to each other whereas AC and BD are non-parallel sides. Also, ‘h’ is the distance between the two parallel sides which demonstrates the height of the trapezium.

## Shape of Trapezium

The shape of trapezium is defined differently in different countries. Based on the sides of the trapezium, there is a confusion in the distinction between the terms “trapezoid” and “trapezium”. Because of the meaning of these two words in US and British, the definition is exactly reversed.

In British, a trapezium defines a quadrilateral with one pair of parallel sides and trapezoid is defined as a quadrilateral with no parallel sides.

In the US, a trapezium has defined as a quadrilateral with no parallel sides and trapezoid is defined as a quadrilateral with one pair of parallel sides.

## Types of Trapezium

The trapezium is further categorised into three different types namely

• Isosceles trapezium – The legs or the non-parallel sides of the trapezium are of equal length
• Scalene Trapezium – A trapezium with all the sides and angles of different measures
• Right Trapezium – A right trapezium has at least two right angles

## Properties of Trapezium

Some of the important properties of a trapezium are as follows:

• In trapezium, exactly one pair of opposite sides are parallel
• The diagonals intersect each other
• The non-parallel sides in the trapezium are unequal except in isosceles trapezium
• The line that joins the mid-points of the non-parallel sides is always parallel to the bases or parallel sides which is equal to half the sum of the parallel sides.

Mid-segment = (AB+ CD)/2

Where AB and CD are the parallel sides or bases

• In isosceles trapezium, the legs or non-parallel sides are congruent
• The sum of the interior angles of the trapezium is equal to 360 degrees. i.e., ∠A + ∠B + ∠C + ∠D = 360°
• The sum of the two adjacent angles is equal to 180°. This means that the two adjacent angles are supplementary.

## Area of Trapezium

The area of a trapezium can be found by taking the average of the two bases of a trapezium and multiply by its altitude. So, the area of trapezium formula is given as

Area of a Trapezium, A = (a+b)/2 square units.

Where,

“a” and “b” are the bases

“h” is the altitude or height

## Perimeter of Trapezium

The perimeter of a trapezium is found by adding all the sides. Therefore, the perimeter of a trapezium formula is given as

The perimeter of Trapezium, P = a + b+ c + d units

Where,

“a, b, c, d” are the sides of the trapezium

## Examples on Trapezium

Q.1: Find the area of the trapezium, in which the sum of the bases(parallel sides) is 60 cm and its height is 20 cm.

Solution: Given;

Let, Sum of the bases, (a+b) = 60 cm

Height,h = 20 cm

We know that,

Area of a Trapezium, A = (a+b)/2 square units.

Substitute the given values,

A = (60/2)× 20

A = 30 × 20

A = 600 cm2.

Therefore, the area of trapezium = 600 cm2.

Q.2: Find the perimeter of a trapezium whose sides are 3cm, 4cm, 5cm and 6cm.

Solution: By the formula of the perimeter of trapezium, we know;

P = Sum of all the sides

P = 3+4+5+6

P = 18cm

Hence, the perimeter is 18cm.

### Does a Trapezium have Parallel Sides?

According to the definition of trapezium, it is a quadrilateral with one pair of parallel sides. Thus, a trapezium does have parallel sides.

### Are the Diagonals of a Trapezium Equal?

No, the diagonals of a trapezium may not be equal. For a trapezium, only 1 pair of its sides are parallel. But, for any quadrilateral to have equal diagonals, two pairs of sides should be parallel like in a square, rectangle, etc.

### Do Diagonals Bisect each other in a Trapezium?

No, the diagonals of a trapezium might not bisect each other. If the diagonals are bisecting, the trapezium will be a parallelogram. So, every parallelogram is a trapezium but every trapezium might not be a parallelogram.

### What is the Difference Between a Trapezium and a Trapezoid?

A trapezium is a geometric figure which has at least one pair of parallel sides. The word “trapezium” is used in the UK, India and other parts outside North America. On the other hand, the word “trapezoid” is used in North America (USA and Canada) and is a geometric figure having at least one pair of parallel sides.