Area of Trapezium

A Trapezium is a type of quadrilateral that has one pair of opposite sides parallel, known as the bases of the trapezium, while the other two sides are non-parallel to each other, also known as the legs. The unique properties of a trapezium, having one pair of parallel sides, make it different from other quadrilaterals like a parallelogram, a rectangle, or a square. The area of a trapezium is an important property of this unique quadrilateral with a wide range of practical and mathematical applications, which we are going to discuss here. 

1.0Area of Trapezium Formula

The area of a trapezium of formula can be defined as the product of half of the sum of the parallel sides (bases) and the perpendicular distance between these sides, that is, the height. 

Mathematically,

The formula to find the area of trapezium can be expressed as: 

$AreaofTrapezium=\frac{1}{2}\times (a+b)\times h$

Here, 

• a​ = length of the first base
• b​ = length of the second base
• h = height (the perpendicular distance between the parallel sides)

2.0Area of Trapezium Formula Without Height

To find the Area of the Trapezium Formula Without Height, suppose you have a trapezium ABCD with sides a, b, c, and d. Suppose the diameter of the trapezium (a line from one corner to the other corner) is denoted as D.

Step 1: Split the Trapezium into Two Triangles

You can divide the trapezium into two triangles by extending a diagonal (the diameter) from one corner of the trapezium to the other. This diagonal cuts the trapezium into two triangles: ΔABD and ΔBCD.

Step 2: Determine the Area of Each Triangle

Now that we have two triangles, we find the area of each triangle individually. You can use the Heron’s formula for the area of a triangle, which is:

$AreaofTriangle=\sqrt{s(s-a)(s-b)(s-c)}$

Here, s is the semi-perimeter of the triangle and can be found using the formula 

$s=\frac{(a+b+c)}{2}$

Step 3: Sum the area of both the triangles 

$AreaofTrapezium=△ABD+△BCD$

3.0Area of Trapezium Derivation 

The area of a trapezium can be derived using two methods: the first is by using the concept of a parallelogram, and the second is by simply summing up the area of all the other shapes within the trapezium. Let’s understand both methods in detail. 

Area of Trapezium and Parallelogram

Suppose you have two identical trapeziums with two parallel sides, say a & b, and the height of the trapezium is h. Now, take the second trapezium and flip it upside down so that when it is placed near one another, it forms a parallelogram, as shown here in the figure. 

Formation of Parallelogram: When we connect the two trapeziums, we have a parallelogram now. The base of the parallelogram is the addition of the two parallel sides of the trapezium, i.e., a+b. The height of the parallelogram is the same as the height of the trapezium, i.e., h.

Area of the Parallelogram: The area of a parallelogram is given by:

Area of Parallelogram= BaseHeight=(a+b)h

As the parallelogram consists of two trapeziums, the parallelogram's area is double the area of one trapezium. Therefore, the area of one trapezium A is half the area of the parallelogram:

$AreaofTrapezium=\frac{1}{2}\times (a+b)\times h$

Sum of the Areas of Other Shapes

According to the figure, the trapezium can be divided into three sub-figures, which include two triangles, say triangle 1 with height h and base a and triangle 2 with height h and base c and one rectangle with sides length b₁ and breath

Hence, the area of the trapezium can be written as the sum of these two triangles and a rectangle. Mathematically, this can be represented as: 

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

we know, Area of Triangle = $\frac{1}{2}$ base height and, Area of rectangle =base height

So, 

$A=\frac{1}{2}\times a\times h+b_1\times h+\frac{1}{2}\times b\times h$

$A=\frac{1}{2}h(a+2b_1+b)$

Or, 

$A=\frac{1}{2}h(a+b_1+b+b_1)$

Here, the longer parallel side of trapezium ABCD is $DC=a+b_1+b$ , so the equation can be written as: 

$A=\frac{1}{2}h(b_1+b_2)$

The above-mentioned formula is the area of the trapezium formula.

4.0Solved Examples: 

Problem 1: Find the area of a trapezium when all sides are given for parallel sides as 20 cm and 8cm and for non-parallel sides as 12 cm and 14 cm. 

Solution:

Given that CD = 8cm, AD = 12cm, BC = 14cm, and AB = 20 cm. 

To find the area of the trapezium given here, we need to find the height CG. To find it, construct a line CF parallel to line AD. 

Here, $\text{CF}\parallel \text{AD and DC}\parallel \text{AF}$

So, DA = CF = 12 cm and AF = DC = 8 cm as opposite sides of a parallelogram are equal. 

Now, using Heron’s formula, find the area of triangle BCF. 

$s=\frac{(12+14+12)}{2}=\frac{38}{2}=19$

$AreaofTriangle=\sqrt{s(s-a)(s-b)(s-c)}$

$AreaofTriangle=\sqrt{19(19-12)(19-12)(19-14)}$

$AreaofTriangle=\sqrt{19(7)(7)(5)}=7\sqrt{95}{\text{ cm}}^2$

The area of a triangle can also be found with: 

$AreaofTriangle=\frac{1}{2}\times BF\times CG$

$7\sqrt{95}=\frac{1}{2}\times 12\times CG$

$CG=\frac{7}{6}\sqrt{95}\text{ cm}$

Now, the area of the Trapezium will be: 

$AreaofTrapezium=\frac{1}{2}\times (a+b)\times h$

$AreaofTrapezium=\frac{1}{2}\times (28)\times \frac{7}{6}\sqrt{95}=159.19{\text{ cm}}^2$

Problem 2: A garden is shaped like a trapezium. The lengths of the two parallel sides are 12 meters and 8 meters, and the height of the garden is 5 meters. Find the area of the trapezium garden.

Solution:

Given a = 12m, b = 8cm, h = 5m 

$AreaofTrapezium=\frac{1}{2}\times (a+b)\times h$

$AreaofTrapezium=\frac{1}{2}\times (12+8)\times 5=50{\text{ m}}^2$

Problem 3: A farmer wants to build a fence around his trapezium-shaped field. The lengths of the parallel sides of the field are 50 meters and 30 meters, and the height of the field is 20 meters. The cost of fencing per meter is INR 50.

Solution:

Given that a = 50m, b = 30m, and h = 20m 

$AreaofTrapezium=\frac{1}{2}\times (a+b)\times h$

$AreaofTrapezium=\frac{1}{2}\times (50+30)\times 20=800{\text{ m}}^2$

The cost of fencing: 80050 = 40,000Rs