The area of a trapezium is the number of unit squares that can fit into the trapezium, measured in square units. Let’s understand in detail the formula of the trapezium area and how to derive it. Area of the trapezium is the region covered by a trapezium in a two-dimensional plane. It is the space enclosed in 2D geometry and measured in square units
What is a Trapezium?
Trapezium, which is also known as the trapezoid, is a closed quadrilateral that contains a pair of parallel sides, whereas the other pair of sides are not parallel. The sides may or may not vary in length.
Parallel sides of the trapezium are called the bases of the trapezium. The distance between the bases is known as the height of the trapezium. The height of the trapezium is also known as the altitude.
The non-parallel sides of the trapezium are known as the legs.
What is the Area of Trapezium?
The area of Trapezium is defined as the region covered by the trapezium in 2-dimensional space. It is measured in square units, for example, meter2, centimeter2, inches2, etc.
The area of the trapezium depends upon its height and the length of its parallel sides.
Illustration of a TrapeziumArea of a trapezium or trapezoid is obtained using parallel lines and the distance between them. In the case of trapezium, the area is given by,
Area = 1/2 × (Sum of Parallel Sides) × (Distance between Parallel Sides)
Let's assume the a and b to be the parallel sides of the trapezium and h to be the distance between them. Let the area of the trapezium be denoted by A. We have,
A = 1/2 × (a + b) × h square units
Perimeter of Trapezium
Perimeter of trapezium is defined as the sum of all the sides of the trapezium. The formula for the perimeter of a trapezium when the sides are "a", "b", "c", and "d" is given by,
Perimeter (P) = (a + b + c + d) units
How to Calculate Area of Trapezium
Area of trapeziumis found using the following the steps added below:
Step 1: Calculate the lengths of the parallel sides (bases) of the trapezium.
Step 2: Sum of the bases of the trapezium is calculated.
Step 3: Value of the sum of bases is multiplied by the height or altitude of the trapezium and then by 1/2.
Step 4: Answer is then further simplified and is written in terms of square units.
Area of Trapezium Formula Derivation
Formula for area of a trapezium can be derived in two different ways. They are:
Trapezium Area Derivation Using a Parallelogram
In order to find the area of a trapezium formula using a parallelogram, we will take two trapeziums that are the same (equal sides and angles), their parallel sides are a and b, and the height of the trapezium is h.
Then we will place the second trapezium upside down. It is clear that in this way if both the trapeziums are joined, it will become a parallelogram.
Derivation of Trapezium Area Using ParallelogramNow, after joining both trapeziums, the trapeziums will form one parallelogram,
Two Trapeziums Forming a ParallelogramLet's say that the area of one trapezium is A, and the area of a parallelogram will be twice the area of the trapezium, that is, 2A. The area of a parallelogram is base × height. Therefore,
2A = Base × Height
Base = (a + b)
Height = h
2A = (a + b) × h
A = 1/2 × (a + b) × h
Area of Trapezium Derivation Using a Triangle
Let us onsider a trapezium with parallel sides as a and b and height as h.
In order to find the area of the trapezium formula using a triangle, bisect the non-parallel side of the triangle. Then we will join it from the corner to make a small triangle, flip the triangle and make it into a bigger triangle.
Derivation of Area of Trapezium FormulaIt is observed that the area of the bigger triangle and the area of the original trapezium are equal. The base of the triangle is (a + b).
Area of triangle is given as,
A = 1/2 × base × height
A = 1/2 × (a + b) × h = Area of Trapezium
Area of Trapezium without Height
Area of the Trapezium can also be calculated even when the height is not given. This is done by following the steps discussed below:
Let us suppose a trapezium ABCD is given with sides as a, b, c, and d respectively, and its diameter is given as D.
Step 1: Divide the given trapezium into two triangles using the diameter as, ΔABD and ΔBCD
Step 2: Calculate the area of triangles ΔABD and ΔBCD separately by using the following Heron's Formula:
Area = √[s⋅(s-a)⋅(s-b)⋅(s-c)]
where,
- a,b, and c are Lengths of Sides of Triangle
- s is Semi-Perimeter of Triangle {s = (a+b+c)/2}
Step 3: Add both the areas of triangles obtained in step 2.
This is required area of Trapezium and it is measured in square units.
Area of Isosceles Trapezium
An isosceles trapezium is a trapezium with congruent base angles and congruent non-parallel sides. The area of an isosceles trapezium or isosceles trapezoid is calculated by multiplying the height of the trapezium by the mean of the parallel sides.
Let us suppose the parallel sides of the isosceles trapezium are "a" and "b" and the height of the trapezium is "h".
Then area of isosceles trapezium is,
A = (a + b)/2 × h
Properties of Trapezium
Properties of trapezium are:
- A trapezium is a two-dimensional figure.
- Bases of the trapezium are parallel to each other.
- Diagonals of an isosceles trapezium are equal in length, and they always intersect each other.
- Sum of the adjacent interior angles is 180°, and the sum of all the interior angles of a trapezium is 360°.
Area of Trapezium Questions
Let us solve some questions on the area of trapezium formula we discussed so far.
Example 1: Find the area of the trapezium with the sum of the parallel sides being 40 m and the height being 20 m.
Solution:
Sum of parallel sides = 40 m
Height of the trapezium = 20 m
As we know that, Area of trapezium = 1/2 × (Sum of the parallel sides) × Height
Area of trapezium = 1/2 × (40) × 20
Area of trapezium = 400 m2
Example 2: Find the sum of the parallel side of the trapezium if its area is 2500 cm2 and height is 50 cm.
Solution:
Here we have to find the sum pf parallel sides of the trapezium
Area of the trapezium = 2500 cm2
Height of the trapezium = 50 cm
As we know that, Area of trapezium = 1/2 × (Sum of the parallel sides) × Height
2500 = 1/2 × (Sum of the parallel sides) × 50
Sum of parallel sides = (2500 × 2) / 50
Sum of parallel sides = 100 cm
Example 3: Calculate the height of a trapezium if the sum of the parallel sides is 200 cm, and the area of the trapezium is 5000 cm2.
Solution:
Here we have to find the height of the trapezium
Area of the trapezium = 5000 cm2
Sum of the parallel sides of the trapezium = 200 cm
As we know that, Area of trapezium = 1/2 × (Sum of the parallel sides) × Height
5000 = 1/2 × 200 × Height
Height = (5000 × 2)/200
Height = 50 cm
Example 4: If the sum of the parallel sides of the trapezium is double the height and the area of the trapezium is 400 m2. Then find the sum of the parallel sides of the trapezium and its height.
Solution:
Area = 400 m2
Sum of the parallel sides of the trapezium is double its height
Let us assume, Height of the trapezium = x, than
Sum of the parallel sides of the trapezium = 2x
As we know that, Area of trapezium = 1/2 × (Sum of the parallel sides) × Height
400 = 1/2 × 2x × x
400 = 1/2 × 2x2
400 = x2
x = √400
x = 20 m
Thus,
Height of the trapezium = x = 20 m
Sum of the parallel sides of the trapezium = 2x = 2 × 20 = 40 m
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