Area of Trapezium – Formulas, Methods & Examples

The area of any 2D plane figure is the region covered by its perimeter or boundary. A trapezium is a type of quadrilateral with only a pair of opposite sides parallel. As there are fixed formulas for finding the area of any 2D figure, the area of a trapezium can be found using certain formulas. 

Let’s understand the various methods and formulas to find the area of trapezium.

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 Area of Trapezium?

The area of a trapezium is the number of unit squares that can fit into the region bounded by its four sides. The area of trapezium is measured in square units such as $m^{2}$, $cm^{2}$, $in^{2}$, $ft^{2}$, etc. Since it’s not always possible to draw an exact number of squares inside the trapezium to find its area, there is a formula to find the area of trapezium.
The formula to find the area of trapezium is “Half the sum of parallel sides multiplied with height”.

For a trapezium having the lengths of the parallel sides as $a$ and $b$ and the height (perpendicular distance between the parallel sides) as $h$, then the formula to calculate the area of trapezium is $\frac {1}{2}\times \left(a + b \right) \times h$.

Derivation of the Area of Trapezium

Consider a trapezium $ABCD$ with parallel sides $BC$ and $DA$ of lengths $a$ and $b$ respectively. Further, let $h$ be the height of the trapezium $ABCD$ (perpendicular distance between the parallel sides).

Let’s flip the trapezium vertically to get a second trapezium (shown green).

Now, let’s combine the two trapeziums to get a parallelogram $PQRS$, having the length of the sides $QR$ and $SP$ equal to $a + b$ and height $h$.

Area of parallelogram is $\text{Base} \times \text{Height}$.

Area of parallelogram $PQRS = \left(a + b \right) \times h$.

As the area of trapezium $ABCD$ is one-half the area of parallelogram $PQRS$, therefore, area of trapezium $ABCD = \frac {1}{2} \times \left(a + b \right) \times h$.

Derivation of the Area of Isosceles Trapezium

Consider an isosceles trapezium $ABCD$, such that $BC || DA$ and non-parallel sides equal,

i.e., $AB = CD$. 

From $A$ and $D$ draw perpendiculars on $BC$ at $E$ and $F$ respectively.

Then $EF = DA = a$.

Let $BE = x$, therefore, $FC = x$, and $x + a + x = b => 2x + a = b => x = \frac {b – a}{2}$.

Area of isosceles trapezium $ABCD$ = (Area of $\triangle ABE$) + (Area of rectangle $AEFD$) + (Area of $\triangle DFC$)

Area of $\triangle ABE$ = Area of $\triangle DFC$ = $\frac {1}{2} \times x \times h = \frac {1}{2} \times \frac {b – a}{2} \times h = \frac {\left( b – a\right) \times h}{4}$.

Area of rectangle $AEFD = a \times h$.

Area of isosceles trapezium $ABCD$ = $\frac {\left( b – a\right) \times h}{4} + a \times h + \frac {\left( b – a\right) \times h}{4} = \frac {\left( b – a\right) \times h}{2} + a \times h $ 

$= \left( \frac {b – a}{2} – a \right) \times h = \frac {b – a + 2a}{2} \times h = \frac {1}{2} \times \left(a + b \right) \times h$.

Note: The formula is the same as that of a non-isosceles trapezium.

Examples

Ex 1: Find the area of a trapezium whose parallel sides are of lengths $8 cm$ and $12 cm$ respectively and the perpendicular distance between the two is $5 cm$.

Lengths of parallel sides of a trapezium are $a = 8 cm$ and $b = 12 cm$

Height of a trapezium $h = 5 cm$

Area of a trapezium = $\frac {1}{2} \times \left(a + b \right) \times h = \frac {1}{2} \times \left(8 + 12 \right) \times 5 = \frac {1}{2} \times 20 \times 5 = 50 cm^{2}$.

Ex 2: The area of a trapezium is $28 m^{2}$ and the length of its parallel sides is $8 m$ and $6 m$ respectively. Find the perpendicular distance between the parallel sides.

Area of trapezium $A = 28 m^{2}$

Length of the parallel sides $a = 8 m$ and $b = 6 m$

Area of a trapezium $A = \frac {1}{2} \times \left(a + b \right) \times h =>  28 = \frac {1}{2} \times \left(8 + 6 \right) \times h => 28 = \frac {1}{2} \times 14 \times h => h = \frac {28}{14} = 2 m$.

Ex 3: The area of a trapezium is $42 cm^{2}$. If the length of one of its parallel sides is $5 cm$ and the height of a trapezium is $6 cm$, then find the length of the second parallel side.

Area of trapezium $A = 42 cm^{2}$

Length of one parallel side $a = 5 cm$

Height of trapezium $h = 6 cm$

Area of a trapezium $A = \frac {1}{2} \times \left(a + b \right) \times h => 42 = \frac {1}{2} \times \left(5 + b \right) \times 6 => 42 = 3 \times \left(5 + b \right) $

$=>5 + b = \frac {42}{3} => 5 + b = 14 => b = 14 – 5 => b = 9 cm$.

The length of the second parallel side of a trapezium is $9 cm$.

Conclusion

A trapezium is a type of quadrilateral having a pair of parallel opposite sides. The area of the trapezium can be calculated by adding the length of the parallel sides and then multiplying half the sum obtained with its height.

Practice Problems

1. Find the area of a trapezium if the length of its two parallel sides is $10 cm$ and $6 cm$ and the perpendicular distance between the two parallel sides is $5 cm$.
2. Find the area of a trapezium if the length of its two parallel sides is $18 cm$ and $12 cm$ and the height is $10 cm$.
3. The area of a trapezium is $100 cm^{2}$. If the length of one of its parallel sides is $12 cm$ and the height of a trapezium is $5 cm$, then find the length of the second parallel side.
4. The area of a trapezium is $140 mm^{2}$ and the length of its parallel sides is $12 mm$ and $10 mm$ respectively. Find the perpendicular distance between the parallel sides.

Recommended Reading

• Area of Rectangle – Definition, Formula & Examples
• Area of Square – Definition, Formula & Examples
• Area of a Triangle – Formulas, Methods & Examples
• Area of a Circle – Formula, Derivation & Examples
• Area of Rhombus – Formulas, Methods & Examples
• Area of A Kite – Formulas, Methods & Examples
• Perimeter of a Polygon(With Formula & Examples)
• Perimeter of Trapezium – Definition, Formula & Examples
• Perimeter of Kite – Definition, Formula & Examples
• Perimeter of Rhombus – Definition, Formula & Examples
• Circumference (Perimeter) of a Circle – Definition, Formula & Examples
• Perimeter of Square – Definition, Formula & Examples
• Perimeter of Rectangle – Definition, Formula & Examples
• Perimeter of Triangle – Definition, Formula & Examples
• What Are 2D Shapes – Names, Definitions & Properties

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