Mensuration is the branch of mathematics that deals with the study of Geometric shapes, their area, volume, and other related parameters. A four-sided closed figure in which one pair of parallel sides opposite to each other and another non-pair of non-parallel sides is called a trapezium.
Properties of Trapezium
- It is a Four-sided Closed Figure with a sum of interior angles 360°.
- One pair of Parallel sides which should be opposite to each other.
- One pair of non-parallel sides.
- Sum of the angles of adjacent sides is 180°.
- Diagonals of a trapezium bisect each other on intersection.
Basic Terminology for Trapezium
Base of a Trapezium: The pair of parallel sides which are opposite to each other are called base. You can call as b1 and b2 respectively.
Height of a Trapezium: The perpendicular distance between the two parallel lines is called as Height of the trapezium.
Formula for Area of Trapezium
If the base and height of a trapezium are given, then the area of a Trapezium can be calculated with the help of the formula:
Area of Trapezium = 1/2 x (sum of bases) x (Height of trapezium)
Derivation for Area of a Trapezium
The area of a trapezium is equal to the sum of the areas of the two triangles and the area of the rectangle. Following is the derivation for calculating the area of the trapezium:
Since we know that:
Area of trapezium = Area of triangle 1 + area of rectangle + Area of triangle 2
Let us suppose the base of triangle 1 be B1 and base of triangle 2 is B2 and the height be h for both the triangles. And for rectangle assume its breadth and height be b1 and h.
That means,
A = (B1 x h / 2) + b1h + (B2 x h / 2)
A = (B1 x h + 2b1h + B2 x h) / 2
Simplifying the equation, and rearranging the terms, and factoring result to:
A = h / 2[b1 + (B1 + b1 + B2] ….(i)
If we assume the longer base of the trapezoid be b2, then
b2 = B1 + b1 + B2 …..(ii)
Substituting (ii) in equation (i),
A = h / 2(b1 + b2)
Therefore, the area of a trapezoid with bases b1, b2, and altitude h is;
A = h/2(b1+b2)
which can also be written as below-
Area of Trapezium = 1 / 2 x (sum of bases) x (Height of trapezium)
= 1 / 2 x ( b1 + b2) x h
Sample Problems based on the Formula
Problem 1: Calculate the area of the trapezium in which the value of bases are 10 and 5 respectively and the height of trapezium is 2 m.
Solution:
Since we know that,
Area of trapezium = 1/2 x (sum of bases) x (Height of trapezium)
= 1/2 x (10 + 5) x 2
= 15 m^2
Problem 2: Given the area of trapezium as 120 m^2 and height of trapezium is 6m and one of base is 4m. Calculate the length of the other base.
Solution:
Area of trapezium = 1/2 x(sum of bases) x ( Height of trapezium)
Let value of other base is b2
Putting all the given values in the above formula, we got
120 = 1/2 x(4+b2) x 6
120 = (4+b2) x3
40 = (4+b2)
b2 = 36 meter
Problem 3: Given the area of trapezium as 220 m^2 and one base is 6m and another base is 4m. Calculate the height of the trapezium.
Solution:
Area of trapezium = 1/2 x (sum of bases) x ( Height of trapezium)
Let value of height be h.
Putting all the given values in the above formula, we got
220 = 1/2 x (4 + 6) x h
220 = 10 x h
22 = h
h = 22m