Perimeter of Trapezium – Definition, Formula & Examples

The perimeter of a plane shape literally means the measurement (meter) around (peri) the boundary of that shape. In other words, you could walk all around the shape, exactly on the boundary, counting your paces as you walk. When you reach your starting point, the number of paces would be the perimeter of the shape. Since the perimeter of any figure is a linear measurement, it is expressed in terms of unit, such as mm, cm, m, ft, yd, in, etc.

Let’s learn what is the perimeter of trapezium and how it is calculated.

Trapezium – A 2D Plane Figure

A trapezium is a type of quadrilateral having a pair of parallel opposite sides. The opposite parallel sides are referred to as the base and the non-parallel sides are referred to as the legs of the trapezium. The figure below shows a trapezium shape.

Note: 

• A quadrilateral is a plane 2D figure with four sides(edges).
• A quadrilateral has $2$ pairs of opposite sides.

In the above figure, $AB$ and $CD$ are the bases while $DA$ and $BC$ are the legs of the trapezium. 

When the length of two legs of a trapezium is equal, it is called an isosceles trapezium. In the figure below, the trapezium $PQRS$ is an isosceles trapezium, where the non-parallel sides(legs) $PQ$ and $RS$ are equal. 

Note: In a trapezium, the parallel sides cannot be equal. If in a quadrilateral parallel sides are equal, it becomes a parallelogram, where both the pairs of opposite sides are parallel.

What is the Perimeter of Trapezium?

The perimeter of a trapezium is the total length of the boundary of the trapezium. Thus, the perimeter of the trapezium is calculated by adding the length of all its sides. Since the perimeter is a linear measurement, it is expressed in linear units like centimetre, metre, feet, inch, kilometre, etc.

For a trapezium $ABCD$ shown above, the perimeter is the sum of its four sides namely, $AB$, $BC$, $CD$, and $DA$.

Perimeter of Trapezium Formula Using Sides

The perimeter of a trapezium with side lengths as $a$, $b$, $c$ and $d$ can be calculated by using the formula Perimeter = $a + b + c + d$, where one pair of values represent lengths of parallel sides and the other pair of values represent lengths of non-parallel sides.

Examples

Ex 1: The parallel sides of a trapezium measure $46 cm$ and $25 cm$. Its other sides are $20 cm$ and $13 cm$. Find its perimeter.

The lengths of the parallel sides $a = 46 cm$ and $c = 25 cm$

The lengths of the non-parallel sides $b = 20 cm$ and $d = 13 cm$.

Perimeter = $a + b + c + d = 46 + 20 + 25 + 13 = 104 cm$.

Ex 2: The lengths of two parallel sides of a trapezium are $5 m$ and $4 m$ respectively. If the non-parallel sides are in the ratio $3:4$ and the perimeter is $23 m$, then find the length of the two non-parallel sides of the trapezium.

The lengths of the parallel sides $a = 5 m$ and $c = 4 m$

Let the lengths of the non-parallel sides be $b = 3x$ and $d = 4x$

Perimeter = $a + b + c + d => 23 = 5 + 3x + 4 + 4x => 23 = 9 + 7x$

$=> 7x = 23 – 9 => 7x = 14 => x = 2$.

Therefore, lengths of the non-parallel sides are $3 \times 2 = 6 m$ and $4 \times 2 = 8 m$.

Perimeter of Trapezium Formula Using Height

The perimeter of a trapezium can be calculated even if the length of one of the sides is missing. In such cases, we use the height of the trapezium (distance between the parallel sides) to get the missing side and then find the perimeter of the trapezium. The missing side can be found by using the Pythagoras theorem as shown in the figure below.

Examples

Ex 1: Find the perimeter of a trapezoid $ABCD$ if its dimensions are given as follows: $AD = 120 m$, $BE = 50 m$, $EF = 120 m$, $FC = 80 m$, $DF = 90 m$.

In the above figure, we see that $AE$ and $DF$ are perpendiculars on $BC$ from $A$ and $D$ at $E$ and $F$ respectively.  Therefore, $AE = FD = 90 m$ and also $EF = DA$. 

Now, $\triangle ABE$ and $\triangle DFC$ are right triangles, right-angled at $B$ and $C$ respectively, i.e. $\angle ABE = \angle DCF =  90^{\circ}$.

So, we can use the Pythagoras theorem, to find the lengths of $AB$ and $CD$.

In $\triangle ABE$, $AB = \sqrt{BE^{2} + AE^{2}} = \sqrt{50^{2} + 90^{2}} = \sqrt{2500 + 8100} = \sqrt{10600} = 102.9 m$

Similarly in $\triangle DFC$, $CD = \sqrt{DF^{2} + FC^{2}} = \sqrt{90^{2} + 80^{2}} = \sqrt{8100 + 6400} = \sqrt{14500} = 120.41 m$

Therefore, perimeter of trapezium is $102.9 + 50 + 120 + 80 + 120.41 + 120 = 593.31 m$.

Conclusion

A trapezium is a type of quadrilateral having a pair of parallel opposite sides. The other pair of sides are non-parallel and are known as legs. The perimeter of a trapezium is calculated by adding the lengths of all four sides.

Practice Problems

1. Find the perimeter of a trapezium with sides $10 cm$, $6 cm$, $8 cm$, and $9 cm$.
2. Calculate the perimeter and the area of a trapezium of height 5 cm if its parallel sides are 4 cm and 10 cm and its non-parallel sides are 6 cm and 8 cm.
3. What is the perimeter of the trapezium in which the sum of lengths of non-parallel sides is $12 in$, and the sum of the parallel sides is $8 in$?

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