Area of a Trapezium: Definition, Formula and Examples

Area of a Trapezium: A trapezium is a type of quadrilateral with one pair of parallel sides. Trapeziums also have a pair of non-parallel sides that are known as legs. The Area of a Trapezium is the space it occupies on a two-dimensional plane. A trapezium has four vertices and four angles. The trapezium area is measured in square units, like; (cm^2, m^2, in^2). To learn more about the trapezium formula, derivation and some examples, read the article.

Definition of a Trapezium

A trapezium is a four-sided polygon with one pair of parallel sides and one pair of non-parallel sides. The trapezium has four vertices and four angles. Parallel sides of the trapezium are bases and non –parallel sides known as legs. A Trapezium is also known as a trapezoid.

Area of Trapezium Formula

Area of a Trapezium \(= {\rm{\;}}\frac{1}{2} \times \left( {{\rm{Sum\;of\;parallel\;sides}}} \right) \times\) Distance between the parallel sides
\(A = \frac{1}{2} \times (a + b) \times h\)
Where, \(a\) and \(b\) are length of parallel sides, \(h\) is the distance between the parallel sides.

Quadrilateral and its Types

A quadrilateral is a two-dimensional shape with four sides, four angles, four vertices. There are different types of quadrilaterals which are square, rectangle, Parallelogram, Trapezium, kite, rhombus, isosceles trapezoid etc.

Square:

All the four sides of a square are equal and all the four angles measure \({90^\circ }.\) Diagonals are bisecting each other. The measure of all the interior angles equals to \({360^ \circ }.\)

Rectangle:

 The opposite sides of a rectangle are equal and each of the angles measures \({\rm{90}}^\circ \) Diagonals are bisecting each other. The measure of all the interior angles equals to \({360^ \circ }.\)

Parallelogram:

The opposite sides are equal and parallel in length. The measure of all the interior angles adds up to \({360^ \circ }.\)

Trapezium:

A Trapezium is having four sides with one pair of parallel sides and one pair of non-parallel sides. The measure of all the interior angles equals to \({360^ \circ }.\)

Rhombus:

In a rhombus all the sides are equal. The rhombus resembles the shape of a diamond. Opposite angles are equal in a rhombus. The measure of all the interior angles equals to \({360^ \circ }.\)

Kite:

Kite is having two pairs of sides with an equal length that are adjacent to each other. The measure of all the interior angles equals to \({360^ \circ }.\)

Isosceles Trapezoid:

An isosceles trapezoid is the same as a trapezium, except the non-parallel sides are equal in length. The measure of all the interior angles equals to \({360^ \circ }.\)

Basic Terms of Trapezium

The Base of a Trapezium: The pair of parallel sides of a trapezium are called bases of a trapezium.

Height of a Trapezium: The perpendicular distance between the parallel sides of a trapezium is known as the height or altitude of the trapezium.

Types of a Trapezium

Scalene Trapezium: In a scalene Trapezium all the sides and angles are of different measures.

Isosceles Trapezium: If both legs of the Trapezium are equal, then the Trapezium is known as isosceles Trapezium.

Right Trapezium: If at least two angles of a Trapezium are right angles, then, the Trapeziums are right Trapeziums.

Properties of a Trapezium

1. Trapezium has four sides. The sides of the trapezium \(ABCD\) are \(AB,\,BC,\,CD,\,DA\)
2. Trapezium is having one pair of parallel and another pair of non-parallel sides. Parallel sides are \(AB\) and \(DC,\) non-parallel sides are \(AD\) and \(BC.\)
3. Trapezium is having \(4\) angles. Four angles of a trapezium \(ABCD\) are Angles \(A,\,B,\,C,\,D.\)
4. The measure of all the four angles is equal to \({360^{\rm{o}}}\)
\(\angle A + \angle B + \angle C + \angle D = {360^{\rm{o}}}\)
5. Two pairs of adjacent angles of a trapezium formed between the two parallel sides and one non-parallel side, sum up to \({180^{\rm{o}}}\)
\(\angle A + \angle D = \angle B + \angle C = {180^{\rm{o}}}\)
6. The legs of the isosceles trapezium are congruent.
7. Non-parallel sides are unequal, except in isosceles trapezium.
8. Parallel sides of the trapezium are bases, and the non-parallel sides are known legs.

Perimeter of a Trapezium

The perimeter of a trapezium is the sum of all the sides of a trapezium. Mathematically it is written as,

Perimeter of a trapezium \(ABCD = AB + BC + CD + DA\)

Derivation of Area of a Trapezium

The area of the trapezium is equal to the sum of the area of two triangles and the area of a rectangle.

\({\rm{Area}}\,{\rm{of}}\,PQRS = {\rm{Area}}\,{\rm{of}}\,\Delta PAS + {\rm{Area}}\,{\rm{of}}\,\Delta QBR + {\rm{Area}}\,{\rm{of}}\,{\rm{Rectangle}}\,ABRS\)
\( = \frac{1}{2} \times PA \times AS + \frac{1}{2} \times QB \times BR + (AB \times BR)\)
\( = \frac{1}{2} \times a \times h + \frac{1}{2} \times b \times h + (x \times h)\)
\( = \frac{{ah}}{2} + \frac{{bh}}{2} + xh = \frac{{ah + bh + 2xh}}{2}\)
\( = \frac{h}{2}(a + b + 2x)\)
\( = \frac{h}{2}(x + (a + b + x))\)
So, the area of the trapezium with bases \({b_1}\) and \({b_2}\) with height \(h\) is,
\(A = \frac{h}{2}\left( {{b_1} + {b_2}} \right)\)

Uses of a Trapezium

A trapezium is a four-sided polygon. A trapezium is having many applications in our daily life. In daily life, we can see the trapezium shapes in the tabletops, bridge supporters, and architectural elements, in windows, doors and in pencil boxes, handbags etc. The concept of the trapezium is used in physics to solve many problems. In mathematics, solving problems based on surface area or for finding the area and perimeter of the complex figures. The formula of the trapezium is used in the construction of the shape of the roof. A trapezium has various applications in day-to-day life.

Summary

A trapezium is a four-sided polygon with one pair of parallel sides and one pair of non-parallel sides. The region occupied by the trapezium on a two – dimensional plane is the area of a trapezium. This article helps to learn about trapezium and its properties, derivation of the area of a trapezium formula. The formula of the trapezium is used in the construction of the shape of the roof.

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Solved Examples

Q.1. The length of two parallel sides of a trapezium is \(6\,{\rm{cm}}\) and \(12\,{\rm{cm}}{\rm{.}}\) If its area is \(63\,{\rm{c}}{{\rm{m}}^2}\) then find the distance between the parallel sides.
Ans: Given: \(a = 6\;\,{\rm{cm}},\;\,b = 12\;\,{\rm{cm}},\;\,A = 63\;\,{\rm{c}}{{\rm{m}}^2},\;h = \)?
Area of a Trapezium \(A = \frac{1}{2} \times (a + b) \times h\)
\( \Rightarrow 63 = \frac{1}{2} \times (6 + 12) \times h\)
\( \Rightarrow 126 = 18 \times h\)
\( \Rightarrow h = \frac{{126}}{{18}} = 7\;\,{\rm{cm}}\)
Therefore, \(7{\rm{\;cm}}\) is the distance between the parallel sides.

Q.2. The length of two parallel sides of a trapezium is \(12\,{\rm{cm}}\) and \(16\,{\rm{cm}}{\rm{.}}\) If its area is \(126\,{\rm{c}}{{\rm{m}}^2},\) then find the distance between the parallel sides.
Ans: Given: \(a = 12\;\,{\rm{cm}},\;\,b = 16\,\;{\rm{cm}},\;\,A = 126\;\,{\rm{c}}{{\rm{m}}^2},\;\,h = \)?
Area of a Trapezium \(A = \frac{1}{2} \times (a + b) \times h\)
\( \Rightarrow 126 = \frac{1}{2} \times (12 + 16) \times h\)
\( \Rightarrow h = 9\,{\rm{cm}}\)
Therefore, \(9{\rm{\;cm}}\) is the distance between the parallel sides.

Q.3. Calculate the area of a trapezium in which the length of the bases is \({\rm{10\;cm}}\) and \({\rm{6\;cm}}\) respectively and the distance between the parallel sides is \({\rm{5\;cm.}}\)
Ans: Given: \(a = 10\;\,{\rm{cm}},\;\,b = 6\;\,{\rm{cm}},\;\,A = \)? \(h = 5\,cm\)
Area of a Trapezium \(A = \frac{1}{2} \times (a + b) \times h\)
\(A = \frac{1}{2} \times (10 + 6) \times 5\)
\( \Rightarrow A = 40\;\,{\rm{c}}{{\rm{m}}^2}\)
Therefore, \(40{\rm{\;c}}{{\rm{m}}^2}\) is the area of the trapezium.

Q.4. Calculate the area of a trapezium whose altitude is \(10\,{\rm{cm,}}\) and the parallel sides are \(14\,{\rm{cm}}\) and \(8\,{\rm{cm}}{\rm{.}}\)
Ans: Given: \(a = 14\;{\rm{cm}},\;\,b = 8\;{\rm{cm}},?,\;\,h = 10\;\,{\rm{cm}}\)
Area of a Trapezium \(A = \frac{1}{2} \times (a + b) \times h\)
\( \Rightarrow A = \frac{1}{2} \times (14 + 8) \times 10\)
\( \Rightarrow A = 110\;\,{\rm{c}}{{\rm{m}}^2}\)
Therefore, \(110{\rm{\;c}}{{\rm{m}}^2}\) is the area of the trapezium.

Q.5. Find the area of the trapezium shown below.

Ans: Given: \(a = 12\;\,{\rm{cm}},\;b = 5\;\,{\rm{cm}},\;\,A = ?,\;\,h = 4\;\,{\rm{cm}}\)
Area of a Trapezium \(A = \frac{1}{2} \times (a + b) \times h\)
\( \Rightarrow A = \frac{1}{2} \times (12 + 5) \times 4\)
\( \Rightarrow A = 34\;\,{\rm{c}}{{\rm{m}}^2}\)
Therefore, \(34{\rm{\;c}}{{\rm{m}}^2}\) is the area of the trapezium.

Q.6. If the length of all the sides of a trapezium is \(4\,{\rm{cm,}}\,12\,{\rm{cm,}}\,{\rm{5}}\,{\rm{cm}}\) and \({\rm{7}}\,{\rm{cm}}{\rm{.}}\) Then, find the perimeter of a trapezium?
Ans: Given: \(a = 4\;\,{\rm{cm}}\,\;,b = 12\;\,{\rm{cm}},\,c = 5\;\,{\rm{cm}},\;\,d = 7\;\,{\rm{cm}}\)
Perimeter of a trapezium \(P = a + b + c + d\)
\( \Rightarrow P = 4 + 12 + 5 + 7\)
\( \Rightarrow P = 28{\rm{\;cm}}\)
Therefore, the perimeter of a trapezium is \({\rm{28}}\,{\rm{cm}}{\rm{.}}\)

Q.7. The length of two parallel sides of a trapezium is \({\rm{24}}\,{\rm{cm}}\) and \({\rm{20}}\,{\rm{cm}}{\rm{.}}\) If its area is \({\rm{330}}\,{\rm{c}}{{\rm{m}}^2},\) then find the distance between the parallel sides.
Ans: Given: \(a = 24\,\;{\rm{cm}},\;\,b = 20\;\,{\rm{cm}},\;\,A = 330\;\,{\rm{c}}{{\rm{m}}^2},\;\,h = \)?
Area of a Trapezium \(A = \frac{1}{2} \times (a + b) \times h\)
\( \Rightarrow 330 = \frac{1}{2} \times (24 + 20) \times h\)
\( \Rightarrow 660 = 44 \times h\)
\( \Rightarrow h = \frac{{660}}{{44}} = 15\)
Therefore, \(15{\rm{\;cm}}\) is the distance between the parallel sides.

Q.8. If the length of all the sides of a trapezium is \(8\,{\rm{cm,}}\,10\,{\rm{cm,}}\,{\rm{5}}\,{\rm{cm}}\) and \({\rm{7}}\,{\rm{cm}}{\rm{.}}\) Then, find the perimeter of a trapezium?
Ans: Given: \(a = 8\,\;{\rm{cm}},\;\,b = 10\;\,{\rm{cm}},\,c = 5\;\,{\rm{cm}},\;\,d = 7\;\,{\rm{cm}}\)
Perimeter of a trapezium \(P = a + b + c + d\)
\( \Rightarrow P = 8 + 10 + 5 + 7\)
\( \Rightarrow P = 30\,\;{\rm{cm}}\)
Therefore, the perimeter of a trapezium is \(30{\rm{\;cm}}.\)

Q.9. Calculate the area of a trapezium whose altitude is \(12\,{\rm{cm,}}\) and the parallel sides are \(14\,{\rm{cm}}\) and \({\rm{10}}\,{\rm{cm}}{\rm{.}}\)
Ans: Given: \(a = 14\;\,{\rm{cm}},\;\,b = 10\;\,{\rm{cm}},\;\,A = ?h = 12\;\,{\rm{cm}}\)
Area of a Trapezium \(A = \frac{1}{2} \times (a + b) \times h\)
\( \Rightarrow A = \frac{1}{2} \times (14 + 10) \times 12\)
\( \Rightarrow A = 144\,\;{\rm{c}}{{\rm{m}}^2}\)
Therefore, \(144{\rm{\;c}}{{\rm{m}}^2}\) is the area of the trapezium.

Q.10. Calculate the area of a trapezium whose parallel sides are \(38.7\,{\rm{cm}}\) and \(22.3\,{\rm{cm}}\) and the distance between them is \(16\,{\rm{cm}}{\rm{.}}\)
Ans: Given: \(a = 38.7\;\,{\rm{cm}},\;\,b = 22.3\;\,{\rm{cm}}\;,\,A = ?,\;\,h = 16\;\,{\rm{cm}}\)
Area of a Trapezium \(A = \frac{1}{2} \times (a + b) \times h\)
\( \Rightarrow A = \frac{1}{2} \times (38.7 + 22.3) \times 16\)
\( \Rightarrow A = 488\,\;{\rm{c}}{{\rm{m}}^2}\)
Therefore, \(488{\rm{\;c}}{{\rm{m}}^2}\) is the area of the trapezium.

FAQs About Area of Trapezium

Let’s look at some of the commonly asked questions about the Area of Trapezium:

Q.1.What is the meaning of the right trapezium?
Ans: A right trapezium is a trapezium having two right angles.

Q.2. What is the area of a trapezium formula?
Ans: Area of a trapezium \({\rm{ = }}\frac{1}{2} \times \left( {{\rm{Sum}}\,{\rm{of}}\,{\rm{parallel}}\,{\rm{sides}}} \right) \times {\rm{Distance}}\,{\rm{between}}\,{\rm{the}}\,{\rm{parallel}}\,{\rm{sides}}\)

Q.3. Does the trapezium have parallel sides?
Ans: Yes, the trapezium is a four-sided figure with one pair of parallel sides.

Q.4.What are the properties of trapezium?
Ans: Some of the properties of a trapezium are,
1. Trapezium has \(4\) sides.
2. A trapezium has pairs of parallel sides and pairs of non-parallel sides.
3. The sum of all the angles of a trapezium \({360^ \circ }.\)

Q.5. Are the diagonals of a trapezium equal?
Ans: No, the diagonals of a trapezium are not equal because all the four sides are not equal in a trapezium.

Q.6. What is the perimeter of a trapezium?
Ans: The perimeter of a trapezium is the sum of all the sides of a trapezium.

Q.7. What is the altitude of a trapezoid?
Ans: The perpendicular distance between the parallel sides of a trapezium is known as the altitude or height of a trapezoid.

Q.8. What is an irregular trapezium?
Ans: The non-parallel sides of a trapezium with unequal lengths is called an irregular trapezium.

Q.9. How do you find the area of a trapezium when given sides?
Ans: Area of trapezium can be solved by using the formula
Area of Trapezium \({\rm{ = }}\frac{{\rm{1}}}{{\rm{2}}}{\rm{ \times (Sum}}\,{\rm{of}}\,{\rm{parallel}}\,{\rm{sides) \times (Sum}}\,{\rm{of}}\,{\rm{parallel}}\,{\rm{sides)}}\) Distance between the parallel sides

Q.10. Does a trapezium have 4 right angles?
Ans: A trapezium can have either two right angles or no right angles at all.

Q.11. How to find the area of a trapezium without height?
Ans: To find the area of a trapezium without height, we need to divide the trapezium into a rectangle and two right triangles.
\({\rm{Area}}\,{\rm{of}}\,{\rm{trapezium}}\,{\rm{ = }}\,{\rm{Area}}\ {\rm{of}}\ ,\Delta \,{\rm{1}}\,{\rm{ + }}\,{\rm{Area}}\,{\rm{of}}\,\Delta \,{\rm{2}}\,{\rm{ + }}\,\,{\rm{Area}}\,{\rm{of}}\,{\rm{Rectangle}}\)
By using the Pythagoras theorem, we can calculate the area of right triangles and the area of a rectangle by using its formula. Then, the sum of the area of two right triangles and one rectangle gives the area of a trapezium.