Trapezium

A quadrilateral is a four-sided polygon with four angles. The sum of its interior angles is always 360 degrees. A trapezium is a specific type of quadrilateral that has at least one pair of parallel sides. Understanding quadrilaterals helps identify the distinct properties of a trapezium and differentiate it from other four-sided shapes.  

 

Table of Content  

• What is a Trapezium?
• Types of Trapezium
• Properties of a Trapezium
• Important Terminologies
• Formula for Area of a Trapezium
• Derivation of Area Formula
• Worked Examples
• Applications of Trapezium in Real Life
• Practice Questions
• Conclusion
• FAQs on Trapezium

 

What is a Trapezium?

Definition:  

A trapezium is a quadrilateral with exactly one pair of parallel sides.  

These parallel sides are called bases.  

The other two sides, which are not parallel, are called legs.  

 

Note:  

In British English, “trapezium” means one pair of parallel sides.  

In American English, “trapezoid” means one pair of parallel sides.  

Key Distinction:  

Unlike a parallelogram, which has two pairs of parallel sides, a trapezium has only one.  

 

Types of Trapezium

a) Scalene Trapezium  

All sides and angles are different lengths and measures.  

There is no symmetry.  

There is only one pair of parallel sides.  

 

b) Isosceles Trapezium  

The legs (non-parallel sides) are equal in length.  

The base angles are equal.  

The diagonals are equal in length.  

It has one line of symmetry.  

 

c) Right-Angled Trapezium  

One leg is perpendicular to the bases.  

It has one or two right angles (90 degrees).  

It is often used in construction and ramps.  

 

Properties of a Trapezium

• A trapezium has exactly one pair of parallel sides (called bases).  

• The non-parallel sides are called legs.  

• The sum of the interior angles is 360 degrees.  

• The diagonals may or may not be equal.  

• In an isosceles trapezium, the diagonals are always equal.  

• The altitude or height is the perpendicular distance between the two bases.  

• The median (midsegment) joins the midpoints of the non-parallel sides and is parallel to the bases.  

 

Important Terminologies

Bases: The two opposite and parallel sides of the trapezium.  

Legs: The non-parallel sides of the trapezium.  

Height (Altitude): The shortest (perpendicular) distance between the two bases.  

Median (Midsegment): A line segment connecting the midpoints of the two legs.  

Median Formula:  

Median = (Base1 + Base2) / 2  

 

Formula for Area of a Trapezium

Standard Area Formula:  

Area = ½ × (Sum of parallel sides) × Height  

Area = ½ × (a + b) × h  

Where:  

a = length of one base  

b = length of the other base  

h = height (perpendicular distance between the bases)  

Explanation:  

The area is the average length of the bases multiplied by the height. It calculates the area as if the shape were an averaged rectangle.  

 

Derivation of Area Formula

Steps:  

• Divide the trapezium into two right triangles and one rectangle, if possible.  

• Use the area formulas of triangles and rectangles to construct the full shape.  

• Alternatively, you can transform the trapezium into a parallelogram or use coordinate geometry.    

 

Worked Examples

Example 1:  Base1 = 10 cm, Base2 = 14 cm, Height = 6 cm  

Area = ½ × (10 + 14) × 6  

= ½ × 24 × 6 = 72 cm²  

 

Example 2:  

Area = 84 cm², Base1 = 12 cm, Base2 = 9 cm  

Find the height.  

Use the formula:  

84 = ½ × (12 + 9) × h  

84 = ½ × 21 × h → 84 = 10.5h → h = 84 / 10.5 = 8 cm  

 

Example 3. Find the area of a trapezium with base₁ = 10 cm, base₂ = 6 cm, and height = 5 cm.

Solution:

We use the formula:
Area = ½ × (base₁ + base₂) × height

Substitute the values:
= ½ × (10 + 6) × 5
= ½ × 16 × 5
= 8 × 5
= 40 cm²

Example 4. In a trapezium, the diagonals intersect, and two perpendiculars are drawn from the non-parallel sides to the longer base (8 cm and 12 cm). The height is the average of the two perpendiculars. The bases are 14 cm and 8 cm. Find the area.

Solution:

Step 1:
Calculate the average height:
= (8 + 12) / 2 = 10 cm

Step 2:
Apply area formula:
Area = ½ × (base₁ + base₂) × height
= ½ × (14 + 8) × 10
= ½ × 22 × 10
= 11 × 10
= 110 cm²

 

Applications of Trapezium in Real Life

• Architecture: Trapeziums are used in sloping roofs, ramps, and support structures because of their stability and shape.  

• Engineering: Beams and brackets may have trapezoidal profiles for load distribution.  

• Design & Art: Trapezoidal shapes appear in graphic design, furniture (like table legs), and decorative tiles.  

• Road & Traffic Design: Road shoulders and signs often use trapezoidal designs.  

• Land Measurement: Irregular plots may be divided into trapeziums for area calculation.  

 

Practice Questions

1. Find the area of a trapezium with bases 10 cm and 12 cm, and height 5 cm.  

2. A trapezium has legs 7 cm and 9 cm, bases 10 cm and 6 cm. The height is 4 cm. Find the area.  

3. The area of a trapezium is 60 cm². Its bases are 10 cm and 8 cm. Find the height.  

4. Prove that the diagonals of an isosceles trapezium are equal using triangle congruence.  

5. A right trapezium has one leg perpendicular to the base with a height of 6 cm. The bases are 5 cm and 9 cm. Find the area.  

 

Conclusion

Understanding the trapezium is essential in geometry. It lays the groundwork for solving complex area, symmetry, and shape problems. Trapeziums have one pair of parallel sides and come in various forms (scalene, isosceles, and right-angled). They frequently appear in academic problems and real-world structures.  

Related links  

Types of Quadrilaterals -Unlock the world of 4-sided shapes! Learn all about the types of quadrilaterals with clear definitions, properties, and visual examples to boost your geometry skills.

Area of Trapezium -Master the formula and techniques to find the area of a trapezium. Includes step-by-step examples, solved problems, and real-life applications for easy learning.

 

Frequently Asked Questions on Trapezium

Q1: What is trapezium and its formula?

Ans: A trapezium is a quadrilateral with at least one pair of opposite sides parallel. These parallel sides are called bases, and the non-parallel sides are called legs.
The formula for the area of a trapezium is:
Area = ½ × (base₁ + base₂) × height

 

Q2: What are the 7 properties of a trapezium?

Ans:

1. It has four sides (is a quadrilateral).

2. It has at least one pair of parallel sides (the bases).

3. The non-parallel sides are called legs.

4. The angles on the same side of a leg are supplementary (add up to 180°).

5. The sum of all interior angles is 360°.

6. In an isosceles trapezium, the legs are equal, and the base angles are also equal.

7. The diagonals may or may not be equal (equal only in isosceles trapezium).
   
   

 

Q3: Does a trapezium have 4 equal sides?

Ans: No, a general trapezium does not have 4 equal sides.
Only a rhombus or a square (which are also quadrilaterals) have 4 equal sides.
An isosceles trapezium can have only the non-parallel sides equal, not all four.

 

Q4: Is a trapezium 180 or 360?

Ans: A trapezium is a four-sided polygon, so the sum of its interior angles is 360 degrees.
Each pair of angles along a leg are supplementary (sum to 180 degrees), but the total across all four angles is always 360°.

 

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