Trapezoid Formula

The area of trapezium is the region covered by a trapezoid in a two-dimensional plane. It is the space enclosed in 2D geometry. A trapezoid is a 2D shape, which falls under the category of quadrilaterals. Trapezoid also has its own properties and trapezoid formula based on area and perimeter.

Trapezoid Formula

What is Trapezoid?

A trapezoid is a quadrilateral, which is defined as a shape with four sides and has only one set of parallel sides. Another name of a trapezoid is trapezium. A trapezium is a type of quadrilaterals, with exactly one pair of parallel sides.

The sides of a trapezoid, which are parallel to each other, are termed as the bases of the trapezoid. The non-parallel sides are known as lateral sides. The distance between the parallel sides is known as the altitude.

Properties of a Trapezoid

In a trapezium, the sum of all the four angles of the trapezium is equal to 360⁰. A Trapezium has 4 unequal sides. A Trapezium has two parallel sides and two non-parallel sides. The diagonals of trapezium bisect each other. The intersection point of the diagonals is collinear to the midpoints of the two opposite sides.

Area of a Trapezium

We can calculate the area of trapezium by using the formula:

Area = \(\frac{1}{2}\)  × (sum of parallel sides) × (distance between them) = \(\frac{1}{2}h (a+b)\)

Where,

h	Distance Between Parallel Lines
a, b	Length of parallel sides

Derivation of Area of a Trapezium

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

We know that

Area of trapezium ABCD = Area of ∆ AED + Area of ∆ BFC + Area of rectangle DEFC

= \((\frac{1}{2}  × AE × DE) + (\frac{1}{2}  × FB ×CF) + (FE ×DE)\)

= \((\frac{1}{2} ×AE × h) + (\frac{1}{2} × FB × h) + (FE ×h)\)

= \(\frac{1}{2}  ×  h  ×(AE + 2FE + FB)\)

= \(\frac{1}{2}  × h  ×  (AE + FE + EB + FE)\)

So,

= \(\frac{1}{2}  ×  h  × (AB + FE)\)

= \(\frac{1}{2}\)  ×  h  ×  (AB + DC) square units.

= \(\frac{1}{2}\)  ×  (sum of parallel sides) ×  (distance between them)

The perimeter of a Trapezium

Perimeter of a trapezium is the sum of all sides of the trapezium.

Perimeter of trapezium PQRS = PQ + RS + QR + RP

Solved Examples

Q.1. The length of the parallel sides of a trapezium is in the ratio 3: 2 and the distance between them is 10 cm. If the area of trapezium is 325 cm², find the length of the parallel sides.

Solution: Let the common ration be x,

Then the two parallel sides are 3x, 2x

Distance between them = 10 cm

Area of trapezium = 325 cm²

Area of trapezium = \(\frac{1}{2}h (a+b)\)

325 = \(\frac{1}{2} (3x + 2x) ×  10\)

i.e. 325 = 5  ×  x  ×   5

⇒ 325 = 25  ͯ  x

⇒ x = \(\frac{325}{25}\)

Therefore, 3  ͯ  x = 3  ͯ  13 = 39 and 2  ͯ  x = 2  ͯ  13 = 26

Length of parallel sides are 26 cm and 39 cm.

Q.2. Find the area of a trapezium with bases of 6 meters and 4 meters and a height of 3 meters.

Solution: Area of trapezium = \(\frac{1}{2}h (a+b)\)

Area = \(\frac{(6+4) ͯ 3}{2}\)

Area = 15 m²