# Trapezoid

Sub Topics
It is a quadrilateral in which one pair of opposite sides are parallel.
In the following trapezoid ABCD, AB // DC and AD and BC are non-parallel sides.

Isosceles trapezoid: A trapezoid in which the non-parallel sides are of equal length is called an isosceles trapezoid. In the isosceles trapezoid, the base angles are equal.
Let us study the following isosceles trapezoid.

In the trapezoid shown above, AB // DC and the non-parallel sides, AD = BC
The base angles are equal.

(i.e) $\angle$DAB = $\angle$ABC and $\angle$ADC = $\angle$BCD

## Properties of Trapezoid

1.    Only one pair of opposite sides are parallel. AB // DC.
2.    The other pair of opposite sides are non-parallel BC is not parallel to AD.
3.    The Diagonals are unequal.
4.    No two angles are equal.
5.    Pairs Interior angles on the same side of the non-parallel sides and between the parallel lines are supplementary.
6.    The sum of all the interior angles = 360°.

Properties of an Isosceles Trapezium:

1.    Only one pair of opposite sides are parallel. AB // DC.
2.    The other pair of opposite sides are non-parallel BC is not parallel to AD.
3.    The Diagonals are equal.
4.    Two pairs of angles are equal.[ pairs of acute angles and pairs of obtuse angles are equal ]
5.    Pairs Interior angles on the same side of the non-parallel sides and between the parallel lines are supplementary.[ sum of an acute + obtuse angle = 180° .
6.    The sum of all the interior angles = 360°.

## Perimeter of Trapezoid

Perimeter of a trapezium is the boundary of the trapezium which is equal to the sum of all the four sides of the trapezium.
In the above trapezium, ABCD, Perimeter = AB + BC + CD + DA
IF AB = a units,  DC = b units and the non-parallel sides BC = c, Ad = d,
then the  Perimeter = a + b + c + d

Example: Find the perimeter of the trapezium bounded by the sides, 10 cm, 5 cm, 8 cm and 6 cm.
Perimeter = 10 + 5 + 8 + 6
= 29 cm

## Area of Trapezoid

Area of a trapezium = $\frac{1}{2}$ [ sum of the parallel sides ] x  height
In the above diagram Area of the Trapezium ABCD = $\frac{1}{2}$ [ AB + CD ] x height

Formula Derivation:
Area of Trapezium ABCD = Area of $\Delta$ABD + Area of $\Delta$BDC

= $\frac{1}{2}$ x AB x h + $\frac{1}{2}$   x CD x h

[ since area of a $\Delta$= $\frac{1}{2}$ x base x height ]

= $\frac{1}{2}$ x h x ( AB + CD )[Factorising by taking $\frac{1}{2}$ h outside ]
Area of Trapezium ABCD    = $\frac{1}{2}$  ( sum of the parallel sides ) x height square units

## Three Dimensional Trapezoid

It is a 3-dimensional solid which has a definite length whose cross section is a trapezium. It has definite length.
For example, the canal in which water flows, swimming pool.

Volume of Trapezoid:

Volume of the trapezoid = Area of cross section x length of the solid

= Area of the trapezium x length (l)

= $\frac{1}{2}$  ( sum of the parallel sides ) x height x length

=  $\frac{1}{2}$  ( a + b ) x h x l  cubic units