Area of Trapezium

Definition of Trapezium

A trapezium is a quadrilateral, meaning it is a polygon with four sides. Among these four sides, only one pair of opposite sides is parallel. These parallel sides are called the bases of the trapezium, while the other two non-parallel sides are known as the legs.

Types of Trapezium

• Isosceles Trapezium: The non-parallel sides (legs) are equal in length.

• Right Trapezium: It has one or two right angles.

To understand the area of a trapezium, you need to recognize its structure. The height (h) of a trapezium is the perpendicular distance between the two parallel sides (the bases).

 

Table of Contents

• What is Area of Trapezium?
• Area of Trapezium Formula
• Derivation of the Area of a Trapezium
  • Area of Trapezium Derivation Using a Parallelogram
  • Area of Trapezium Derivation Using a Triangle
• How To Find Area of Trapezium?
• Area of Trapezium by Heron’s Formula
• Properties of a Trapezium
• Applications of Area of Trapezium
• Solved Examples of Area of Trapezium
• Conclusion
• FAQs On Area of Trapezium

 

What is Area of Trapezium?

So, what is the area of  trapezium?

The area of trapezium is the amount of surface enclosed within its four sides. It is measured in square units such as cm², m², ft², etc. Since a trapezium is not a regular polygon like a square or rectangle, its area is calculated using a specific formula that considers the lengths of both parallel sides and the height.

 

Area of Trapezium Formula

The formula for the area of a trapezium comes from the average of the lengths of the two parallel sides, multiplied by the height:  

Area of a Trapezium = 1/2 × (a + b) × h  

Where:  

• a and b are the lengths of the parallel sides (also called bases)  

• h is the height, which is the straight distance between the bases.  

This formula helps us easily find the area of a trapezium, even when the sides are different lengths.  

Example:

If a trapezium has bases 10 cm and 6 cm, and the height is 4 cm:
Area = 1/2 × (10 + 6) × 4
          = 1/2 × 16 × 4 = 32 cm²

This simple calculation highlights how to find the area of trapezium with known base lengths and height.

 

Derivation of the Area of a Trapezium  

Understanding how to derive the area of a trapezium helps students see why the formula works, not just how to use it.  

There are two standard ways to derive the formula:  

• Using a parallelogram  

• Using triangles  

 

Area of Trapezium Derivation Using a Parallelogram  

Visualize a trapezium with two bases, a and b, and height, h. Now, copy the trapezium and rotate it to create a parallelogram by placing the two trapeziums side by side.  

• The combined base of the parallelogram equals a + b.  

• The height remains h.  

• The area of the parallelogram equals (a + b) × h.  

Since this figure is made of two trapeziums, the area of one trapezium is:

Area = 1/2 × (a + b) × h

This derivation confirms the area of a trapezium formula and explains its origin using known shapes.

 

Area of Trapezium Derivation Using a Triangle

Another way to find the area of a trapezium is by splitting it into two triangles. You can do this using one of the diagonals or a line from the non-parallel sides.  

Each triangle has:  

• Base = one of the parallel sides (either a or b)  

• Height = the common height h  

So the total area is:  

Area = 1/2 × h × a + 1/2 × h × b = 1/2 × (a + b) × h  

Once again, we get the same trapezium formula. This shows students that no matter how we break it down, the formula stays the same.  

 

How To Find Area of Trapezium?

To find the area of trapezium, follow these easy steps:

1. Measure the length of both parallel sides. Let’s call them a and b.

2. Measure the height h, which must be perpendicular to both bases.

3. Plug these values into the area of a trapezium formula:

Area = 1/2 × (a + b) × h

Example:

Given:

• a = 14 cm

• b = 10 cm

• h = 6 cm

Area = 1/2 × (14 + 10) × 6
        = 1/2 × 24 × 6
        = 72 cm²

This is a direct method to find the area of trapezium when dimensions are available.

 

Area of Trapezium by Heron’s Formula

Sometimes, the height isn't provided, but you may have all four side lengths of the trapezium. In these cases, divide the trapezium into two triangles and apply Heron's Formula.

Heron's Formula:  

For a triangle with sides a, b, c:  

s = (a + b + c)/2  

Area = √[s(s - a)(s - b)(s - c]  

Divide the trapezium diagonally, apply Heron’s Formula to each triangle, and add both triangle areas to find the total area of the trapezium. This method is particularly useful when it's hard to measure the height directly.

 

Properties of a Trapezium

• One pair of opposite sides is parallel (bases).

• The non-parallel sides are called legs.

• The angles on the same side of a leg are supplementary.

• If the non-parallel sides are equal, it's an isosceles trapezium.

• The diagonals usually intersect, and their lengths are generally unequal.

• The area of a trapezium depends on both base lengths and height.

Understanding these properties gives more insight into why the area of trapezium formula works the way it does.

 

Applications of Area of Trapezium

The area of a trapezium is used in various real-life applications and professions:  

• Land Measurement: Many plots of land have a trapezoidal shape.  

• Construction and Architecture: Sloped roofs, bridges, and trusses often have trapezium shapes.  

• Interior Design: Furniture like desks and benches are sometimes trapezoidal.  

• Road Engineering: Cross-sections of roads or embankments are trapeziums.  

• Irrigation Channels: Many cross-sections of canals are trapezium-shaped.  

In these fields, professionals often calculate the area of a trapezium to determine material quantities, dimensions, or cost estimates.

 

Solved Examples of Area of Trapezium

Example 1:

Given: a = 20 cm, b = 10 cm, h = 6 cm
Find: Area of a trapezium
Solution:
Area = 1/2 × (20 + 10) × 6 = 1/2 × 30 × 6 = 90 cm²

 

Example 2:

Given: Bases = 30 m and 20 m, height = 10 m
Find: Area
Area = 1/2 × (30 + 20) × 10 = 250 m²

 

Example 3:

Given:

• Base1 = 18 cm

• Base2 = 12 cm

• Height = 7 cm

Find: Area of trapezium

Solution:
Area = 1/2 × (18 + 12) × 7
Area = 1/2 × 30 × 7
Area = 15 × 7 = 105 cm²

 

Example 4:

Given:

• Base1 = 25 m

• Base2 = 15 m

• Height = 10 m

Find: Area of trapezium

Solution:
Area = 1/2 × (25 + 15) × 10
Area = 1/2 × 40 × 10
Area = 20 × 10 = 200 m²

 

Example 5:

Given:

• Base1 = 40 mm

• Base2 = 30 mm

• Height = 12 mm

Find: Area of trapezium

Solution:
Area = 1/2 × (40 + 30) × 12
Area = 1/2 × 70 × 12
Area = 35 × 12 = 420 mm²

 

Example 6:

Given:

• Base1 = 60 cm

• Base2 = 40 cm

• Height = 15 cm

Find: Area of trapezium

Solution:
Area = 1/2 × (60 + 40) × 15
Area = 1/2 × 100 × 15
Area = 50 × 15 = 750 cm²

 

Conclusion

The area of a trapezium is an important concept in geometry. It has many real-world uses, from school problems to building projects. Whether you're dealing with land plots or exam questions, knowing how to calculate the area of a trapezium using the trapezium formula or Heron’s Formula is important.

 

Related Links

Area of a Triangle - Understand how to calculate the area of a triangle with easy methods.

Area and Perimeter - Learn how to find the area and perimeter of different shapes with examples.

 

Frequently Asked Questions on Area of Trapezium

1. What is the area of the trapezium formula?

Ans: The area of the trapezium formula is:
Area = 1/2 × (a + b) × h
Where:

• a and b are the lengths of the two parallel sides (bases)

• h is the height (perpendicular distance between the bases)

This formula calculates the total surface area enclosed by a trapezium.

 

2. Why is the area of a trapezium?

Ans: The area of a trapezium is calculated to determine the amount of space enclosed within its four sides. It is derived by taking the average of the two bases (parallel sides) and multiplying it by the height. This works because a trapezium can be broken into simpler shapes like triangles or rearranged into a parallelogram.

 

3. What are all the formulas of trapezium?

Ans: Here are the main formulas related to a trapezium:

• Area of a trapezium:
  Area = 1/2 × (a + b) × h

• Perimeter of a trapezium:
  Perimeter = a + b + c + d
  Where a and b are bases, c and d are non-parallel sides (legs)

• Length of mid-segment (median):
  Median = (a + b) / 2

These formulas help in solving most geometry problems involving a trapezium.

 

4. What is the area of the trapezium?

Ans: The area of the trapezium is the amount of 2D space enclosed within the four sides of the trapezium. It is measured in square units like cm², m², etc., and can be found using the formula:
Area = 1/2 × (base1 + base2) × height

 

5. How to find the area of a trapezoid?

Ans: To find the area of a trapezoid (another word for trapezium in US English), follow these steps:

1. Measure the lengths of the two parallel sides (base1 and base2)

2. Measure the height (the perpendicular distance between the bases)

3. Plug into the formula:
   Area = 1/2 × (base1 + base2) × height

This will give you the total area of the trapezoid in square units.

 

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