Area of Trapezium

Trapezium or trapezoid is a quadrilateral consisting of one pair of parallel sides. It is a 2D structure with four uneven sides and angles. An area of any shape can be defined as the region enclosed by the boundaries of that particular shape.

A trapezium can be judged based on the following properties

It has one pair of parallel sides and another pair of non-parallel sides.

The sum of two adjacent angles of a trapezium is 180°.

Sum of all the interior angles of a trapezium is equal to, just like any other quadrilateral, 360°.

Diagonals of a trapezium bisect each other.

The length of mid-segment is equal to half of the sum of the lengths of two parallel bases.

Before we move on to find out the Area of a trapezium, let’s explore some of the basic terminologies that will be useful to proceed further.

Basic terminology

Base: The two parallel sides of a trapezium are called base regardless of how the trapezium is being placed. They can be of the same or different lengths. Non- parallel sides of the trapezium are called legs.

Height: The maximum distance between the two parallel sides of the trapezium is called the height of the trapezium. 

Mid-segment: The line segment drawn from the midpoint of one non-parallel side to another is called the mid-segment. It is usually parallel to the bases of the trapezium and divides it into two equal parts.

Isosceles trapezium: This is a special type of trapezium in which the lengths of non-parallel sides are equal.

Area of trapezium

Consider a trapezium of parallel sides of lengths ‘a’ and ‘b’ such that ( a<b ) and it’s height be of length ‘h.’ Place the trapezium such that ‘b’ is the base. 

Draw perpendiculars from the ends of side ‘a’ on ‘b’ such that the trapezium gets divided into two right-angled triangles of height ‘h’ and base (b-a)/2 each and one rectangle of length and breadth ‘a’ and ‘h’ respectively. 

Now place the triangles joined height to height to make a scalene or an isosceles triangle as the resulting figure. This triangle will be of height ‘h’ and base (b-a). 

On finding the sum of areas of the rectangle and the resultant triangle we get, 

Area of trapezium = Area of rectangle + Area of resultant 

                                                                    triangle

Area of trapezium = ( a×h ) + { ½ × (b-a) ×h }

                             = h { a + ½b – ½a }

                             = h { ½a + ½b }

                             = h { ½ ( a+b ) }

                             = ½ h ( a+b )

Therefore, 

Area of trapezium is equals to ( ½ × distance between the parallel sides × sum of parallel sides )

Examples of the trapezium in daily life

Trapezoid shape can be seen in the architectural designs and projection drawings in engineering.

In daily life, we can see a trapezium shape in certain types of windows and doors and tabletops. 

Also, our pencil boxes sometimes resemble the shape of a trapezium. 

If you ever look closely at the side view of your purse, you’ll find a trapezoid-shaped gap between the front face and the back end of your purse.

/ Exam, General

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About the Author

Rachit believes in the power of education and has studied from the top institutes of IIIT Allahabad, IIM Calcutta, and Francois Rabelias in France. He has worked as Software Developer with Microsoft and Adobe. Post his MBA, he worked with the world’s # 1 consulting firm, The Boston Consulting Group across multiple geographies US, South-East Asia and Europe.

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