Area of Trapezium Formula: Definition, Derivation, Properties, Examples

The Area of Trapezium formula represents the two-dimensional space enclosed by this specific geometric shape. In the realm of 2D geometry, we quantify this space in square units. A trapezium is classified as a quadrilateral, belonging to the category of two-dimensional shapes. Much like other geometric figures, trapeziums possess distinct properties and formulas governing their area and perimeter. This article elucidates the formula for calculating the area of a trapezium, offering insights into its derivation and providing illustrative examples for enhanced comprehension.

What is Area of Trapezium?

A trapezium, classified as a quadrilateral with one pair of parallel sides, encompasses a region within its four sides. The area of a trapezium hinges on the lengths of these parallel sides and the height of the figure, and it is quantified in square units. Let's delve into the formula for determining the area of any trapezium:

Area of a Trapezium Formula

The area of a trapezium can be computed using the following formula:

Area = (1/2) × h × (a + b)

Where:

a and b represent the lengths of the parallel sides or bases of the trapezium.

h signifies the height or the distance between the parallel sides.

Perimeter of Trapezium

The perimeter of a trapezium, which is the total length of its boundary or the sum of its four sides, can be calculated using the following formula:

Perimeter =  a+b+c+d

Where:

a and b are the lengths of the two parallel sides (also known as bases) of the trapezium.

c and d are the lengths of the other two non-parallel sides (also known as the legs) of the trapezium.

Simply add the lengths of all four sides together to find the perimeter of the trapezium.

Also Check – Conic Section Formula

Derivation of Area of a Trapezium

To find the area of a trapezium, we break it down into the sum of three components: two triangles and one rectangle.

We can express this as:

Area of the trapezium = Area of Triangle 1 + Area of Rectangle + Area of Triangle 2.

In mathematical terms:

A = (1/2 * a * h) + (b1 * h) + (1/2 * c * h),

Which simplifies to:

A = (ah + 2b ₁ h + ch) / 2.

Further simplification, rearranging terms, and factoring gives us:

A = h/2[b ₁ + (a + b ₁ + c)] ….(i)

Now, if we designate the longer base of the trapezium as b2, we can express it as:

b2 = a + b ₁ + c ....(ii).

By substituting equation (ii) into equation (i), we arrive at the final formula for the area of a trapezium with bases b1 and b2 and altitude h:

A = h/2(b ₁ + b ₂ )

Also Check – Linear Inequalities formula

How to Calculate Area of Trapezium?

To calculate the area of a trapezium, you can use the following steps:

Step 1: Determine the dimensions of the given trapezium, including the lengths of its parallel sides and the height.

Step 2: Add the lengths of the parallel sides together.

Step 3: Multiply the sum of the parallel sides by the height of the trapezium.

Step 4: Finally, multiply the result from step 3 by 1/2 to obtain the final answer.

The value obtained in step 4 represents the area of the given trapezium.

Trapezium Basic Concept

• In a trapezium, the pair of sides that are parallel to each other are known as the bases, while the sides that are not parallel are referred to as the legs.
• The mid-segment is a line segment that connects the midpoints of the non-parallel sides of a trapezium.
• Take a look at the various types of trapezium figures shown above, where the arrow indicates the parallel sides. In all three figures, you'll notice that two sides are parallel to each other, while the other two sides are not.
• If we draw a line segment between the midpoints of the two non-parallel sides, it effectively divides the trapezium into two unequal parts.
• Similar to isosceles triangles, where two sides of a triangle are equal, and the angles opposite those equal sides are also equal, there exists a figure known as an Isosceles Trapezium. In an Isosceles Trapezium, the two non-parallel sides are equal in length and form equal angles with one of the bases. You can see an example of this in the third figure provided above.

Properties of a Trapezium

Here, we will explore additional characteristics of the trapezium, which is also known as a trapezoid. Trapezium exhibits the following properties:

• Similar to other quadrilaterals, the sum of all four angles in a trapezium amounts to 360°.
• A trapezium possesses two sides that are parallel to each other, and two sides that are not parallel.
• In a regular trapezium, the diagonals intersect and bisect each other.
• The length of the mid-segment, which connects the midpoints of the parallel bases, is equal to half the sum of the lengths of those parallel bases.
• Two pairs of adjacent angles within a trapezium, formed between the parallel sides and one of the non-parallel sides, combine to form a total of 180 degrees.

Also Check – Complex Number Formula

Applications of Area of Trapezium Formula

This concept finds extensive application in various physics calculations and mathematical computations. It serves as the fundamental framework for deriving equations of motion, as elaborated in the 9th CBSE science textbook. The integration of physics equations with mathematical calculations is presented in a manner that enhances the comprehension of aspiring engineering students.

Solved Examples on Area of Trapezium Formula

Example 1:

Let's find the area of a trapezium with parallel sides of lengths 6 cm and 8 cm and a height of 5 cm.

Solution: Given,

length of parallel sides of trapezium;

a = 6cm

b = 8cm

Height of trapezium, h = 5cm

Therefore,

area of trapezium = 1/2 (a+b).h

= 1/2 (6 +8) x 5

=35 sq cm.

Example 2:

Calculate the area of a trapezium with parallel sides of lengths 10 cm and 15 cm and a height of 8 cm.

Given,

length of parallel sides of trapezium;

a = 10cm

b = 15cm

Height of trapezium, h = 8cm

Therefore,

area of trapezium = 1/2 (a+b).h

= 1/2 (10 +15) x8

=100 sq cm.