A quadrilateral is a two-dimensional closed shape which has four sides, four corners, and four vertices. According to Euclidean Geometry, a quadrilateral is a polygon having 4 sides, 4 vertices. There are seven quadrilaterals and they are:

Square: It has all sides equal and makes at the edges. Diagonal intersects each other. The sum of all the interior angles is.

Rectangle: Pair of sides are equal and all the edges are at. Diagonals intersect each other. Sum of interior angles is .

Rhombus: It is similar to a square. All sides are equal. It is of the shape of the diamond. Opposite angles are equal. Sum of interior angles is

Trapezium: It has a pair of parallel sides. Sum of interior angles is

Kite: Two pairs of equal-length sides that are adjacent to each other. Sum of interior angles is

Parallelogram: Opposite sides are parallel and equal in length. Sum of interior angles is

Isosceles Trapezoid: It is also the same as trapezium. The only difference is non-parallel sides are of equal length. Sum of interior angles is

This article focuses on Trapezium, its basic concept, its properties, area, application, formulae and derivation of the area of trapezium.

Definition:

The trapezium is a two-dimensional closed figure having a pair of parallel sides. It has 4 sides and 4 vertices. Parallel sides of the trapezium are bases and non-parallel sides are called legs.

Scalene Trapezium: All the sides and angle are of different measures. Just trapezoid is scalene trapezium as shown in the figure below.

Isosceles Trapezium: If in trapezium any two pair of sides are equal, that is, bases or legs then the trapezium is Isosceles.

Right Trapezium: At least two of the angles are right angle i.e.,

The basic difference between the trapezium and trapezoid is shown below. This difference is only due to British and American versions.

Bases are the parallel sides of the trapezium and non-parallel sides are the legs.

A line drawn from the mid of non- parallel sides is the midpoint.

The arrows and equal marks shown in the figure denotes that the lines are parallel and the length of the sides are equal respectively.

The trapezium will get divided into two unequal parts if one cut it into two sides from the mid of non-parallel sides.

In Isosceles trapezium, the two non-parallel sides are equal and form equal angles on the bases.

Some of the properties of Trapezium are as follows -

Exactly one pair of opposite sides are parallel.

Diagonals intersect each other.

The sum of the internal angles of the trapezium is i.e.,

Except for isosceles trapezium, trapezium has non-parallel sides unequal.

The line that joins the mid-point of the non-parallel sides is always parallel to the bases of the trapezium.

Mid-segment = (AB+ CD)/2

The legs are congruent in Isosceles Trapezium.

The two angles of a trapezium are supplementary to each other. Their sum is equal to.

The area of trapezium is calculated as it is half of the sum of parallel sides and height. Mathematically it is written as :

The perimeter of the trapezium is the sum of all the four sides. Mathematically it is given as,

Perimeter = AB + BC + CD + DA

Derivation of the Area of Trapezium :

The derivation of the area of trapezium is given below.

To Derive: Area of trapezium

Derivation:

Here, let one side be ‘b1’ and other side be ‘b2’

Distance between the parallel sides is ‘h’

From the figure, it can be seen that there are two triangles and one rectangle.

Hence, the area of trapezium is

Area = area of triangle 1 + area of rectangle + area of triangle 2

Area = ++

=

=

On simplifying the above equation, we get,

=

From the figure, we know

= b2 = a+b1+b

Therefore,

Area =

Substituting the value from the above equation we get,

=

=

is the required equation.

Hence, the area of the trapezium has been derived where b1 and b2 are bases and h is the altitude.

The concept of trapezium has a wide range of applications. It is used in physics for solving various questions based on trapezium whereas in mathematics it is used in variety of applications, i.e, solving various questions based on surface area or for finding the complex figure area or perimeter. Trapezium formula can be used in construction also like the shape of the roof is trapezoidal. It has multiple applications in daily life.

Example:

The length of the parallel sides of a trapezium are in the ratio 5:2 and the distance between them is 20 cm. If the area of trapezium is 325 cm², find the length of the parallel sides.

Solution:

Let x = common ratio

The parallel sides are = 5x and 2x

Altitude = 20cm

Area of trapezium = 325cm2

Then from the formula of area of trapezium

Area=

325 = x(5x+2x)x20

= x = 4.64cm

Hence, the parallel sides are

5x = 23.2cm

2x = 9.28cm

Therefore, the length of the non - parallel sides is 23.2cm and 9.28cm.

Example:

Two parallel sides of a trapezium are of lengths 15 cm and 10 cm respectively, and the distance between them is 22 cm. Find the area of the trapezium.

Solution:

Given :

Parallel sides of the trapezium = 15 cm and 10 cm

Distance between the parallel sides = 22 cm

Area of trapezium is given by =

Hence,

Area= (15+10) x 22 = 275 cm2

Therefore, the area of the trapezium is 275 cm2.

Example:

Find the perimeter and area of the trapezium whose parallel sides are 10 cm and 25 cm. The distance between the bases is 30 cm and the non-parallel side length is 20 cm each.

Solution:

Given :

Parallel sides length = 10 cm and 25 cm

Altitude = 30 cm

Area=

= (10+25) x 30

= 1050 cm2

Perimeter= Sum of all the sides

= 10+25+20+20

=75 cm

Hence, the area and perimeter is 1050 cm2 and 75 cm respectively.