Properties of Trapezium: Definition, Formula, Examples

A Trapezium is a convex quadrilateral with two parallel sides of opposite lengths and two unequal sides. Let us dive into the world of learning and understanding the Properties of Trapezium. People usually rely on different shapes for building, creating, or measuring anything. One of the most common shapes is Trapezium; It is a two-dimensional shape representing a table when drawn on paper. The trapezium pattern can be seen in objects like a glass of water, a table lamp, a flower pot, and boats.

One of the main applications of the Trapezium is the trapezium rule, where the area under the curve is divided into various numbers of trapeziums. When considering multiple applications like construction, interior design, animation, 3D printing, and other areas of work, it is essential to properly measure the size, shape, volume, and other aspects of a shape. In this article, we will learn the types of trapeziums, such as irregular trapeziums, and even check the solved examples for knowing how to solve practical problems.

In this article, students will be able to learn all about Trapezium. Read on to discover all the various aspects of Trapezium and how to solve problems associated with them.

What is Trapezium?

A Trapezium is a convex quadrilateral with at least one pair of parallel sides. The parallel sides are called bases, and the other two sides are called the legs or the lateral sides.

A trapezium is a \(4\)-sided flat shape with straight sides that have a pair of parallel opposite sides. 

1. Bases are the parallel sides and non-parallel sides are known as legs in a trapezium.
2. A line drawn from the mid of non-parallel sides is called the mid-segment.
3. In an isosceles trapezium, the \(2\) non-parallel sides are equal and form equal angles with the bases.
4. The trapezium will get divided into \(2\) unequal parts if one cuts it into \(2\) sides from the mid of the non-parallel sides.

Shape of Trapezium

A trapezium is a closed shape with four sides. One pair of sides are equal, whereas the other pair is not.

The above figure shows a quadrilateral \(ABCD\) in which opposite sides \(AB\) and \(DC\) are parallel to each other whereas the opposite sides \(AD\) and \(BC\) are not parallel to each other, therefore quadrilateral \(ABCD\) is a trapezium.

As we have learned that the sides \(AB\) and \(CD\) are parallel sides in a trapezium, we get two pairs of co-interior angles. One pair is \(\angle A\) and \(\angle D\) whereas the other pair is \(\angle B\) and \(\angle C.\) The sum of each pair of co-interior angles is \(180^\circ .\) And, therefore, \(\angle A + \angle D = 180^\circ \) and \(\angle B + \angle C = 180^\circ .\)

Types of Trapezium

Based on the sides of the angles, trapeziums can be classified into three types.

1. Isosceles Trapezium

If the non-parallel sides of a trapezium are equal, it is called an isosceles trapezium.

The above figure shows a trapezium \(ABCD\) in which non-parallel sides \(AD\) and \(BC\) are of equal length. Hence, the given trapezium is an isosceles triangle.

Properties of an Isosceles Trapezium

In case of an Isosceles trapezium \(ABCD\) (as shown below)

1. \(\angle A = \angle B\) and \(\angle C = \angle D.\)
2. \(\angle A + \angle D = 180^\circ \) and \(\angle B + \angle C = 180^\circ .\) 
3. \(\angle A + \angle C = 180^\circ \) and \(\angle B + \angle D = 180^\circ .\)
4. Diagonal \(AC = \) Diagonal \(BD \).

2. Right Trapezium

A right trapezium is a trapezium that has two adjacent right angles.

Here, \(\angle A = \angle D = 90^\circ \)

3. Acute Trapezium

An acute trapezium has two adjacent acute angles on its longer base edge while an obtuse trapezium has one acute and one obtuse angle on each base.

Properties of Trapezium

The term trapezium in Greek means “a little table”. Its properties are as follows:

1. It is a polygon.
2. It is a two-dimensional figure like a rectangle, square and parallelogram.
3. The diagonals of a trapezium intersect each other.
4. A trapezium has supplementary adjacent angles.
5. Except for an isosceles trapezium, a trapezium has non-parallel sides unequal.
6. Only one pair of opposite sides are parallel.
7. Since it is a quadrilateral, the sum of internal angles is equal to \(180^o.\)
8. The line that joins the mid-point of the non-parallel sides is always parallel to the bases.
9. The mid-points of sides are collinear to the intersection point of diagonals.

Trapezium Area and Perimeter

A trapezium’s area can be calculated with the following formula:

\({\rm{Area}}\,{\rm{of}}\,{\rm{a}}\,{\rm{trapezium}} = \frac{1}{2}\, \times \,\,{\rm{Sum}}\,{\rm{of}}\,{\rm{parallel}}\, {\rm{sides}}\, \times \,{\rm{Distance}}\,{\rm{between}}\,{\rm{the}}\,{\rm{parallel}}\,{\rm{sides}}\)

Let \({b_1}\) and \({b_2}\) be the lengths of these bases. The distance between the bases is called the height of the Trapezium. Let \(h\) be the height. Then the formula becomes: \(A = \frac{1}{2}\left( {{b_1} + {b_2}} \right) \times h\)

Let’s work out how we landed on this formula.

Given a trapezium, let ​\({b_1}\) and \({b_2}\) be the lengths of the bases, and let \(h\) be the height.

Now, draw a segment (as shown in the below figure) parallel to the bases that are halfway between the bases. This line segment divides the trapezium into two, each with the same height of \(\frac{1}{2}h.\)

Labelling the Angles of the Trapezium

Now, the bases of the trapezium are parallel.
Thus,

\(\angle 4 + \angle 5 = 180^\circ \) and \(\angle 1 + \angle 7 = 180^\circ \) \(\angle 2 = \angle 6\) and \(\angle 3 = \angle 8\)

Now flip the top of the trapezium and place it adjacent to the bottom trapezium as in the following figure:

Now, if we look closely, the shape obtained is in the shape of a parallelogram. The length of its base is \(\left( {{b_1} + {b_2}} \right),\) and its height is \(\frac{1}{2}h.\)
We know that area of a parallelogram \( = base\, \times \,height\) \( = \left( {{b_1} + {b_2}} \right) \times \frac{1}{2}h\)
And hence the area of a trapezium \(\frac{1}{2} = \left( {{b_1} + {b_2}} \right) \times h.\)

Perimeter of the Trapezium

In geometry, the perimeter can be defined as the path or the boundary that surrounds a shape. It can also be defined as the length of the outline of a shape. A trapezium is no exception here. Hence, the perimeter of a trapezium is the sum of the lengths of its four sides.

The four sides of the given trapezium \(ABCD\) are \(AB,\,BC,\,CD\) and \(AD.\)

Perimeter of the trapezium \( = AB + BC + CD + AD\)

Trapezium and Trapezoid

The trapezium is a four-sided polygon and a two-dimensional figure with exactly one pair of parallel sides opposite each other.

On the other side, a trapezoid is also a four-sided polygon with no pair of parallel sides opposite each other.

There is always a conflict between the US version of trapezium or Trapezoid and the British version.

What is an Irregular Trapezium?

A trapezium has one pair of parallel sides, and the other two sides are non-parallel. Now a regular trapezium will have the other two non-parallel sides equal, whereas an irregular trapezium will have the other two non-parallel opposite sides, unequal.

How to Find the Angle and Unit Area of Trapezium?

For a regular or isosceles trapezium, the sets of angles adjoined by parallel lines are equal. Also, we know, for any quadrilateral the sum of all the interior angles is equal to \(360^\circ .\)

Hence, if an angle, say \(y,\) is given between one parallel side and one non-parallel side, then subtracting twice of this angle from \(180^\circ \) will give the sum of \(2\) angles on the formed opposite side of \(y.\) Once, the sum is determined, the measure of the fourth angle can be obtained by dividing the sum by \(2.\)

The area of the trapezium is the amount of space it occupies or encloses in the plane. The area is usually measured in square units like square metres, square feet, square inches, etc.

Solved Examples

Question-1: If the sum of the parallel sides of a trapezium \(8\,{\rm{cm}}\) and the distance between them is \(5\,{\rm{cm,}}\) then what is the area of the Trapezium?
Answer: The area of a trapezium is the mean of the lengths of the parallel sides times the distance between the parallel sides.

In this case, the sum of the length of the parallel sides is \(8\,{\rm{cm}}.\)
The distance between the parallel sides is \(5\,{\rm{cm}}.\)

\( \Rightarrow \) The area of a trapezium is \(\frac{1}{2}\left( {{b_1} + {b_2}} \right) \times h = \frac{1}{2} \times 8 \times 5 = 20\,{\rm{c}}{{\rm{m}}^{\rm{2}}}\)

Question-2: Find the area of the Trapezium, in which the sum of the parallel sides is \(80\,{\rm{cm,}}\) and its height is \(20\,{\rm{cm}}{\rm{.}}\)
Answer: Given, the sum of parallel sides \(80\,{\rm{cm}}\) and height, \(h = 20\,{\rm{cm}}\)
We know that,

Area of a trapezium, \(A = \frac{1}{2} \times {\rm{Sum}}\,{\rm{of}}\,{\rm{parallel}}\,{\rm{sides}} \times {\rm{Distance}}\,{\rm{between}}\,{\rm{the}}\,{\rm{parallel}}\,{\rm{sides}}\)

Substitute the given values,
\({\rm{A}} = \frac{1}{2} \times 80 \times 20\)
\({\rm{A}} = 40 \times 20\)
\({\rm{A}} = 800\,{\rm{c}}{{\rm{m}}^{\rm{2}}}.\)

Therefore, the area of trapezium \( = 800\,{\rm{c}}{{\rm{m}}^{\rm{2}}}.\)

Question-3: The length of the parallel sides of a trapezium are in the ratio \(3:2\) and the distance between them is \(10\,{\rm{cm}}.\) If the area of Trapezium is \(425\,{\rm{c}}{{\rm{m}}^{\rm{2}}},\) find the length of the parallel sides.
Answer: Let the common multiple be \(y,\)

Then the two parallel sides are \(3y\) and \(2y\)
Distance between them \( = 10\,{\rm{cm}}\)

Area of trapezium \( = 425\,{\rm{c}}{{\rm{m}}^{\rm{2}}}\)
Area of a trapezium \( = \frac{1}{2} \times {\rm{Sum}}\,{\rm{of}}\,{\rm{parallel}}\,{\rm{sides}} \times {\rm{Distance}}\,{\rm{between}}\,{\rm{the}}\,{\rm{parallel}}\,{\rm{sides}}\)
\(425 = \frac{1}{2}\left( {3y + 2y} \right)10\)
\( \Rightarrow 425 = 5y \times 5\)
\( \Rightarrow 425 = 25y\)
\( \Rightarrow y = \frac{{425}}{{25}} = 17\)

Therefore, \(3y = 3 \times 17 = 51\) and \(2y = 2 \times 17 = 34\)

Hence, the length of parallel sides are \(51\,{\rm{cm}}\) and \(34\,{\rm{cm}}{\rm{.}}\)

Question-4: Find the perimeter of a trapezium whose sides are \(5\,{\rm{cm}},\,6\,{\rm{cm}},\,7\,{\rm{cm}}\) and \({\rm{8}}\,{\rm{cm}}{\rm{.}}\)
Answer: \({\rm{The}}\,{\rm{perimeter}} = {\rm{Sum}}\,{\rm{of}}\,{\rm{the}}\,{\rm{length}}\,{\rm{of}}\,{\rm{all}}\,{\rm{the}}\,{\rm{sides}}\)

Therefore,
\({\rm{P}} = 5\,{\rm{cm}} + 6\,{\rm{cm}} + 7\,{\rm{cm}} + 8\,{\rm{cm}} \)
\({\rm{P}} = 26\,{\rm{cm}}\)

Hence, the perimeter is \(26\,{\rm{cm}}{\rm{.}}\)

Question-5: If a base angle of a trapezium, included between a parallel side and a non-parallel side is equal to \(70^\circ .\) Then find the angle opposite to the given angle.
Answer: Given, one of the base angles of a trapezium is \(70^\circ .\)

Let the angle to be found \( = y\)
By the angle sum property of a quadrilateral, the sum of all the angles equal to \(360^\circ .\)
Twice of given angle \( = 2 \times 70^\circ = 140^\circ \)
Now subtract \(140^\circ \) from \(360^\circ ,\)
\(360^\circ – 140^\circ = 220^\circ \)

Hence, \(220^\circ \) is the sum of the above two angles.
Since, both the above angles are equal for an isosceles trapezium.

Hence,
\(y = \frac{{220^\circ }}{2} = 110^\circ \)

Related Articles to Study

In this article, we explored the whole new world of a trapezium. We learned what a trapezium is and how to find a trapezium area. We also came across the different types of the trapezium and learned what makes them unique in their way.

We also learned where we call a trapezium a trapezoid, its definition, and its features.

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