Formula and Properties of Area of Trapezium
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
Bases are the parallel sides of the trapezium and non-parallel sides are the legs.
A line drawn from the middle of non-parallel sides is the midpoint.
The arrows and equal marks shown in the figure denotes that the lines are parallel and the length of the sides are equal respectively.
The trapezium will get divided into two unequal parts if one cuts it into two sides from the middle of non-parallel sides.
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.
FAQs on Area of Trapezium
1. Are the diagonals of a trapezium equal and do they bisect each other?
A trapezium is a quadrilateral with one pair of parallel sides. However, for the diagonals of a quadrilateral to be equal, two pairs of the figure would have to be parallel as is the case in a square or a rectangle. Thus, the diagonals of a trapezium are not equal but they always bisect each other. In this article too, under the sub-heading ‘Properties of A Trapezium’, we have stated how the diagonals of such a figure always bisect each other.
2. What is the difference between a trapezium and a trapezoid?
We all know that US and UK English differ in a few aspects. Both a trapezium and a trapezoid roughly refer to the same geometrical figure that has four sides, four vertices, and four angles. As per the British convention, a trapezoid is a quadrilateral in which no sides are parallel whereas a trapezium is a quadrilateral in which one pair of sides are parallel. In contrast to this, the opposite is true in the case of the convention followed in America. Indian textbooks followed by the CBSE, ICSE, and state board curriculums usually refer to this Euclidean figure as a trapezium and not a trapezoid.
3. How can young students distinguish between a trapezium, a parallelogram, and a kite?
Young students can find it difficult to differentiate between these 3 geometrical shapes as they look quite similar. To be able to identify each of them correctly, you must be aware of the properties of each figure. A parallelogram is a quadrilateral that has two sides parallel to each other. As a consequence of this, opposite sides and angles of a parallelogram turn out to be equal. The area of a parallelogram relies both on its base and on its height. The trapezium is also another shape that has two of its sides parallel to each other. These parallel sides are referred to as bases whereas the non-parallel sides of a trapezium are called legs. On the other hand, a kite is a quadrilateral with two pairs of adjacent and congruent sides.
4. What is an isosceles trapezium?
We have discussed the various types of trapeziums in detail in this article. An isosceles trapezium is one such type of trapezium. The simplest way to identify this figure is by looking at the non-parallel sides of the given trapezium. If both the legs or the non-parallel sides of the figure are equal to each other, the trapezium is an isosceles one. The angles opposite to those two equal sides will also turn out to be equal in this case which makes our calculations easier during the time of the examination. Knowing these little tips and tricks can be extremely beneficial and save students a lot of crucial time during their final papers.