Area of a Trapezium: Formula, Definition, Examples, and Practice Questions

Area of a Trapezium: In geometry, a quadrilateral with at least one pair of parallel sides is commonly known as a trapezoid in American and Canadian English, while it is referred to as a trapezium in British and other international English variants. A trapezium is a convex quadrilateral with exactly one pair of parallel opposite sides, known as bases, and one pair of non-parallel sides, known as legs. By definition, a quadrilateral is a polygon featuring four vertices and four sides.

In the geometric figure, sides AB and CD represent the parallel bases, while AD and BC are the non-parallel legs, with 'h' representing the perpendicular altitude between the bases. This article explores the trapezium definition, various types of trapezoids, the area formula, the perimeter calculation, essential properties, and practical solved problems.  

Types of Trapezium 

There are three distinct categories of trapeziums based on their side properties:

1. Isosceles Trapezium
2. Right Trapezium
3. Scalene Trapezium

1. Isosceles Trapezium: A trapezium where the non-parallel legs are of equal length, and the base angles are congruent.

2. Right Trapezium: A trapezium that contains two adjacent right angles (90 degrees).

3. Scalene Trapezium: A trapezium with no sides of equal length.

Area of Trapezium

The area of a shape is the 2D space enclosed within its boundary, measured in square units (e.g., cm², m²). Understanding the properties and formulas of a trapezium is essential for geometry. Below, we explain the derivation of the area formula and provide examples to simplify the concept.

Let us break down the mathematical formula used to calculate the area of a trapezium.

Area of Trapezium  Formula

You can calculate the area of a trapezium using the following formula:

Area of a Trapezium = ½ × (Sum of parallel sides) × (Distance between them)

Area of a Trapezium = ½ × h × (a + b)

Where:

“a” and “b” represent the lengths of the two parallel bases, and 

“h” denotes the height, or the perpendicular distance between the parallel sides.

Perimeter of Trapezium 

The perimeter is the total length of the boundary of a closed shape. It is calculated by summing the lengths of all four sides and is expressed in linear units like meters or centimeters.

The perimeter of a trapezium is determined using this formula:

Perimeter of a Trapezium = Sum of all sides = AB + BC + CD + DA

Properties of Trapezium   

Every trapezium adheres to the following properties:

1. One pair of opposite sides are parallel.
2. The two non-parallel sides are unequal except for the isosceles Trapezium.
3. Two pairs of adjacent angles add up to 180 degrees.
4. The diagonals intersect.
5. The line that joins the mid-points of the non-parallel sides is always parallel to the bases or parallel sides which is equal to half the sum of the parallel sides
6. The sum of the four interior angles is 4 right angles, i.e. 3600.
7. The sum of the four exterior angles is 4 right angles, i.e. 3600.

Uses of a Trapezium 

A trapezium is a four-sided polygon with diverse real-world applications. Common examples include tabletop designs, architectural bridge supports, window frames, doors, and various household objects like handbags. These concepts are vital in physics for problem-solving and in mathematics for determining surface area and complex geometric volumes.

How to derive the formula to find an area of the Trapezium?

Follow this derivation to understand how we reach the area formula for a trapezium:

Area of Trapezium ABCD = Area of ΔABD + Area of ΔCBD

= (½ × a × h) + (½ × b × h)

= ½ × h × (a + b)

= ½ (Sum of parallel sides) × (Perpendicular distance between them)

 

 

Step by step process is given below to calculate the area of the Trapezium:-

Step 1: Identify the dimensions of the trapezium, specifically the lengths of the two parallel bases and the height.

Step 2: Calculate the sum of the parallel sides.

Step 3: Multiply this sum by the height of the trapezium.

Step 4: Multiply the result by ½ to find the area.

The result from Step 4 is the total area of the trapezium.

Use the following examples to practice calculating the area of a trapezium.

Area of Trapezium Related Questions

Below are practice questions to help clarify these concepts.

Question 1: Calculate the area of a trapezium with parallel sides of 24 cm and 20 cm, and a distance of 15 cm between them.

Solution:

Given:

a = 24 cm

b = 20 cm

Height (h) = 15 cm

Area = ½ × (a+b) × h

= {½ × (24+20) × 15} cm²

= (½ × 44 × 15) cm²

= (22 × 15) cm²

= 330 cm²

The area of the trapezium is 330 cm².

Question 2: Calculate the area of a trapezium where the bases are 12 cm and 20 cm, and the altitude is 10 cm.

Solution: 

Given:

a = 12 cm

b = 20 cm

h = 10 cm

Area = ½ × (a+b) × h

= {½ × (12+20) × 10} cm²

= (½ × 32 × 10) cm²

= (16 × 10) cm²

= 160 cm²

The area of the trapezium is 160 cm².

Question 3: The area of a trapezium is 352 cm² and the height is 16 cm. If one parallel side measures 25 cm, find the length of the other base.

Solution: 

Let the unknown base be x cm.

Area = {½ × (25 + x) × 16} cm²

352 = (200 + 8x)

352 - 200 = 8x

152 = 8x

x = 152 / 8

x = 19 cm

The length of the other side is 19 cm.

 

Question 4: The parallel sides of a trapezium are in the ratio 3:2 and the height is 8 cm. If the area is 400 cm², find the lengths of the parallel sides.

Solution:

Let the parallel sides be 3x and 2x.

Formula: Area = ½ × (3x + 2x) × 8

400 = ½ × (5x) × 8

400 = 20x

x = 20 cm. Therefore, the sides are 60 cm and 40 cm.

Practice Trapezium Questions

1. Find the area of a Trapezium whose lengths of parallel sides are 10 cm and 6cm, respectively and the distance between the parallel sides is 5 cm.
2. If the area of a Trapezium is 728 cm2. Find the height of the Trapezium such that the lengths of its parallel sides are 16 cm and 7 cm, respectively.
3. The area of a Trapezium is 384cm². If 3:5 is the ratio of the length of its parallel sides and the perpendicular distance between them is 12 cm. Find the length of each of the parallel sides.