Area of a Trapezoid

Geometry has many uses and one of it is that it provides accurate formulas for calculating the areas and volumes of different geometrical shapes. Geometry plays an important role in engineering sciences and especially in mechanical engineering. So, if you are taking your first mathematics course and you are wondering why you need to learn to calculate the area of a trapezoid, it is because it will be useful later when you move over to applied sciences.

What is a Trapezoid?

A trapezoid (also known as 'trapezium' in British English) is a quadrilateral (closed four sided geometrical figure) that has one pair of parallel sides. Some mathematicians also define a trapezoid as a quadrilateral that has minimum one pair of parallel sides. This second alternative definition, makes parallelograms, rhombuses, squares and rectangles to be special cases of trapezoids.

The parallel sides of a trapezoid may be unequal in length and the rest of the two sides may not be parallel. If the two non-parallel sides of the trapezoid are equal in length, it is called an isosceles trapezoid.

Trapezoid Properties

The base angles of an isosceles trapezoid are equal. The non-parallel sides of such a trapezoid are equal by definition. A necessary and sufficient condition for any quadrilateral to be called a trapezoid is that it must have two adjacent angles, whose sum is 180^o. That means, every trapezoid has to have a pair of supplementary angles. Another interesting property of a trapezoid is that a line drawn, joining the mid points of the parallel sides of every trapezoid, divides it into two parts with equal areas.

Calculation

There are two main formulas that can serve the purpose of calculating trapezoid area. One uses the height of the trapezoid as a parameter along with the lengths of the parallel sides, while the other one gives the area by using just the lengths of sides as parameters.

Trapezoid Area Formula (With Height)
Let us see what is the formula for a trapezoid, when its height is known.

Area = ½ x Height x (Sum of Lengths of Parallel Sides) = ½ H (a + b)

where 'H' is the height of the trapezoid, while 'a' and 'b' are the lengths of its parallel sides. The unit of area will be cm² or m².

Trapezoid Area Formula (Without Height)
What if the height of the trapezium is not known or provided in the problem? What do you do then? In such a case, there is another formula:

Area = [(a + b) / (b-a)] x √[(s - b) (s - a) (s - b - c) (s - b - d)]

where a, b are the lengths of the parallel sides, c, d are the length of the other two sides and s is the semi-perimeter of the trapezoid given by (s = (a + b + c + d) / 2).

Calculating area of any geometrical object is just a matter of plugging in numbers. Find out the derivation of the formulas stated above and derive the area equation of a trapezoid once. It will acquaint you with the method of geometric derivations and besides that, it will be a good exercise for your brain!