[Maths Class Notes] on Area of Trapezium Pdf for Exam

A quadrilateral is a two-dimensional closed shape that has four sides, four corners, and four vertices. According to Euclidean Geometry, a quadrilateral is a polygon having 4 sides, 4 vertices. There are seven quadrilaterals and they are:

Square:

It has all sides equal and makes 90° at the edges. Diagonals intersect each other. The sum of all the interior angles is ∠A + ∠B + ∠C + ∠D = 360°.

Rectangle: 

The pair of sides are equal and all the edges are at 90°. Diagonals intersect each other. Sum of interior angles is ∠A + ∠B + ∠C + ∠D = 360°.

Rhombus:

It is similar to a square. All sides are equal. It is in the shape of a diamond. Opposite angles are equal. Sum of interior angles is ∠A + ∠B + ∠C + ∠D = 360°.

Trapezium:

It has a pair of parallel sides. Sum of interior angles is ∠A + ∠B + ∠C + ∠D = 360°.

Kite: 

Two pairs of equal-length sides that are adjacent to each other. Sum of interior angles is ∠A + ∠B + ∠C + ∠D = 360°.

Parallelogram: 

Opposite sides are parallel and equal in length. Sum of interior angles is ∠A + ∠B + ∠C + ∠D = 360°.

Isosceles Trapezoid:

It is also the same as trapezium. The only difference is non-parallel sides are of equal length. Sum of interior angles is ∠A + ∠B + ∠C + ∠D = 360°.

This article focuses on Trapezium, its basic concept, its properties, area, application, formulae, and derivation of the area of trapezium. 

Trapezium Definition

The trapezium is a two-dimensional closed figure having a pair of parallel sides. It has 4 sides and 4 vertices. Parallel sides of the trapezium are bases and non-parallel sides are called legs. 

Different Types of a Trapezium

Scalene Trapezium:

All the sides and angles are of different measures. Just trapezoid is scalene trapezium as shown in the figure below.

  

(Images will be uploaded soon).

Isosceles Trapezium: 

If in trapezium any two pairs of sides are equal, that is, bases or legs then the trapezium is Isosceles. 

Right Trapezium: 

At least two of the angles are right angles i.e.,90°.

The basic difference between the trapezium and trapezoid is shown below. This difference is only due to British and American versions. 

Difference Between Trapezium and Trapezoid

British Trapezoid:

A quadrilateral with no sides parallel

USA Trapezium:

A quadrilateral with no sides parallel. 

Basic Concepts of Trapezium

1. Bases are the parallel sides of the trapezium and non-parallel sides are the legs.

2. A line drawn from the middle of non-parallel sides is the midpoint.

3. The arrows and equal marks shown in the figure denotes that the lines are parallel and the length of the sides are equal respectively.

4. The trapezium will get divided into two unequal parts if one cuts it into two sides from the middle of non-parallel sides.

5. In Isosceles trapezium, the two non-parallel sides are equal and form equal angles on the bases. 

Properties of Trapezium

Some of the properties of Trapezium are as follows:

Exactly one pair of opposite sides are parallel.

• Diagonals intersect each other. 

• The sum of the internal angles of the trapezium is 360° i.e., ∠A + ∠B + ∠C + ∠D = 360°.

• Except for isosceles trapezium, trapezium has non-parallel sides unequal.

• The line that joins the mid-point of the non-parallel sides is always parallel to the bases of the trapezium. Mid-segment = (AB + CD)/2 

• The legs are congruent in Isosceles Trapezium.

• The two angles of a trapezium are supplementary to each other. Their sum is equal to 180°.

Area of Trapezium

The area of trapezium is calculated as it is half of the sum of parallel sides and height. The formula of area of Trapezium is written as ½ × sum of parallel sides × times distance between them = ½ × (b₁ × b₂) × h

The Perimeter of Trapezium 

The perimeter of the trapezium is the sum of all four sides. Mathematically it is given as, Perimeter = AB + BC + CD + DA. 

Derivation of the Area of Trapezium

The derivation of the area of trapezium is given below. To Derive: Area of trapezium

Derivation: Here, let one side be ‘b1’ and another side be ‘b2’. The distance between the parallel sides is ‘h’

From the figure, it can be seen that there are two triangles and one rectangle. Hence, the area of the trapezium is

Area = area of triangle 1 + area of rectangle + area of triangle

Area = ½ × AE × DE + DE × EF + ½ × FB × CF

          = [frac{ah}{2} + b_{1} h + frac{ch}{2}]

          = [frac{ah+2b_{1} h+ch}{2}]

On simplifying the equation, we get, 

          = [frac{h}{2}] (a + 2b[_{1}] + c)

From the figure, b[_{2}] = a + b[_{1}] + c

Substituting the value from the above equation we get,

          = [frac{h}{2}] [[b_{1}+b_{2}]]

          = [frac{1}{2}] [[b_{1}+b_{2}]] × h

is the required equation. Hence, the area of the trapezium has been derived where b₁ and b₂ are bases and h is the altitude.

Hence, the area of the trapezium has been derived where b₁ and b₂ are bases and h is the altitude. 

How To Find The Angles Of A Trapezium

We now know that there are several types of trapeziums. Keeping a regular or isosceles trapezium in mind, students should remember that the sets of angles adjoined by parallel lines are equal. Moreover, we also know the mathematical fact that the sum of all the interior angles is equal to 360 degrees in any given quadrilateral. This will help us to calculate the angle of any trapezium.

For example, let us assume that an angle named x is given between the two parallel sides and non-parallel sides of the trapezium provided. Now, subtract twice of this angle from 360. This will give you the sum of two angles on the formed opposite side of the angle we named x. Once you have successfully calculated the sum of these two angles, you can proceed to find the fourth angle by dividing this sum by 2. 

Application of Trapezium

The concept of trapezium has a wide range of applications. It is used in physics for solving various questions based on trapezium whereas in mathematics it is used in a variety of applications, i.e, solving various questions based on surface area or for finding the complex figure area or perimeter. Trapezium formula can be used in construction also like the shape of the roof is trapezoidal. It has multiple applications in daily life. 

Example:

The length of the parallel sides of a trapezium are in the ratio 5:2 and the distance between them is 20 cm. If the area of the trapezium is 325 cm², find the length of the parallel sides.

Solution:

Let x = common ratio

The parallel sides are = 5x and 2x

Altitude = 20cm

Area of trapezium = 325cm²

Then from the formula of area of trapezium

Area = [frac{1}{2}] [[b_{1}+b_{2}]] × h

325  = [frac{1}{2}] [5x + 2x] × 20

         = x = 4.64 cm

Hence, the parallel sides are 5x = 23.2 cm and 2x = 9.28 cm

Therefore, the length of the non – parallel sides is 23.2 cm and 9.28 cm. 

Example:

Two parallel sides of a trapezium are of lengths 15 cm and 10 cm respectively, and the distance between them is 22 cm. Find the area of the trapezium. 

Solution:

Given: Parallel sides of the trapezium = 15 cm and 10 cm

Distance between the parallel sides = 22 cm

Area of trapezium is given by = [frac{1}{2}] [[b_{1}+b_{2}]] × h

Hence,

Area = [frac{1}{2}] (15 + 10) × 22 = 275 cm²

Therefore, the area of the trapezium is 275 cm². 

Example:

Find the perimeter and area of the trapezium whose parallel sides are 10 cm and 25 cm. The distance between the bases is 30 cm and the non-parallel side length is 20 cm each.

Solution:

 

Given: Parallel sides length = 10 cm and 25 cm

Altitude = 30 cm

Area = [frac{1}{2}] [[b_{1}+b_{2}]] × h

         = [frac{1}{2}] (10 + 25) × 30

         = 1050 cm²

Perimeter = Sum of all the sides= 10 + 25 + 20 + 20= 75 cm

Hence, the area and perimeter is 1050 cm² and 75 cm respectively.