Properties of Trapezium
A trapezium (called "trapezoid" in American English) is a quadrilateral in which exactly one pair of opposite sides is parallel. The parallel sides are called bases and the non-parallel sides are called legs.
The trapezium is different from a parallelogram — a parallelogram has BOTH pairs of opposite sides parallel, while a trapezium has only ONE pair parallel. This distinction is important in understanding the hierarchy of quadrilaterals.
In Class 8, you will study the basic properties of a trapezium, the special case called the isosceles trapezium, and how to calculate its area. The trapezium appears in many real-life structures — the cross-section of dams, certain table tops, and bucket shapes are all trapezoidal.
What is Properties of Trapezium?
Definition: A trapezium is a quadrilateral in which one pair of opposite sides is parallel.
In trapezium ABCD:
- AB || CD (these are the bases; AB is the longer base and CD is the shorter base, or vice versa)
- AD and BC are not parallel (these are the legs)
Key Terms:
- Parallel sides = bases of the trapezium
- Non-parallel sides = legs of the trapezium
- Height (h) = perpendicular distance between the two bases
Important: A trapezium is NOT a parallelogram. If both pairs of opposite sides become parallel, it is a parallelogram, not a trapezium.
Properties of Trapezium Formula
Area of a Trapezium:
Area = (1/2) x (a + b) x h
Where:
- a = length of one parallel side (base)
- b = length of the other parallel side (base)
- h = height (perpendicular distance between the parallel sides)
Perimeter of a Trapezium:
Perimeter = a + b + c + d
Where a, b, c, d are the four sides. There is no shortcut formula since all four sides can be different.
Derivation and Proof
Deriving the area formula:
Consider trapezium ABCD with AB || CD, where AB = a (bottom base) and CD = b (top base), and the height = h.
Method 1: Using triangles
- Draw diagonal AC. This divides the trapezium into two triangles: triangle ACD and triangle ABC.
- Triangle ABC has base AB = a and height = h. Its area = (1/2) x a x h.
- Triangle ACD has base CD = b and height = h. Its area = (1/2) x b x h.
- Area of trapezium = (1/2) x a x h + (1/2) x b x h = (1/2) x (a + b) x h.
Method 2: Using a parallelogram
- Take two identical trapeziums ABCD.
- Flip one and join it to the other along a leg.
- The combined shape is a parallelogram with base (a + b) and height h.
- Area of parallelogram = (a + b) x h.
- Area of one trapezium = (1/2) x (a + b) x h.
Types and Properties
Trapeziums can be classified into the following types:
1. Scalene Trapezium (General Trapezium):
- The two legs are of different lengths.
- No special symmetry.
- The non-parallel sides and angles are unequal.
2. Isosceles Trapezium:
- The two legs are equal in length.
- The base angles are equal (angles on the same base are equal).
- The diagonals are equal in length.
- It has one line of symmetry (the perpendicular bisector of the bases).
- The non-parallel sides make equal angles with each base.
3. Right Trapezium:
- One of the legs is perpendicular to the bases.
- Two adjacent angles are 90 degrees.
- The perpendicular leg serves as the height of the trapezium.
Properties common to ALL trapeziums:
- Exactly one pair of sides is parallel.
- The sum of all interior angles is 360 degrees.
- Co-interior angles (angles between a parallel side and a leg on the same side) are supplementary — they add up to 180 degrees.
- The diagonals generally do NOT bisect each other.
Solved Examples
Example 1: Example 1: Finding the area
Problem: A trapezium has parallel sides of 12 cm and 8 cm, and a height of 5 cm. Find its area.
Solution:
Given:
- a = 12 cm, b = 8 cm, h = 5 cm
Using the formula:
- Area = (1/2) x (a + b) x h
- Area = (1/2) x (12 + 8) x 5
- Area = (1/2) x 20 x 5
- Area = 50 cm²
Answer: The area is 50 cm².
Example 2: Example 2: Finding the height
Problem: The area of a trapezium is 84 cm². The parallel sides are 10 cm and 14 cm. Find the height.
Solution:
Given:
- Area = 84 cm², a = 10 cm, b = 14 cm
Using the formula:
- 84 = (1/2) x (10 + 14) x h
- 84 = (1/2) x 24 x h
- 84 = 12h
- h = 84/12 = 7 cm
Answer: The height is 7 cm.
Example 3: Example 3: Finding a parallel side
Problem: A trapezium has area 120 cm², height 8 cm, and one parallel side 18 cm. Find the other parallel side.
Solution:
Given:
- Area = 120 cm², h = 8 cm, a = 18 cm
Using the formula:
- 120 = (1/2) x (18 + b) x 8
- 120 = 4 x (18 + b)
- 30 = 18 + b
- b = 12 cm
Answer: The other parallel side is 12 cm.
Example 4: Example 4: Co-interior angles
Problem: In trapezium ABCD, AB || CD. Angle A = 65 degrees. Find angle D.
Solution:
Given:
- AB || CD, angle A = 65 degrees
Angles A and D are co-interior angles (on the same side, between the parallel lines AB and CD).
Co-interior angles are supplementary:
- angle A + angle D = 180 degrees
- 65 + angle D = 180
- angle D = 115 degrees
Answer: angle D = 115 degrees.
Example 5: Example 5: All angles of a trapezium
Problem: In trapezium PQRS, PQ || SR. Angle P = 70 degrees and angle Q = 110 degrees. Find angles R and S.
Solution:
Given:
- PQ || SR, angle P = 70 degrees, angle Q = 110 degrees
Using co-interior angle property:
- angle P + angle S = 180 (co-interior angles)
- angle S = 180 - 70 = 110 degrees
- angle Q + angle R = 180 (co-interior angles)
- angle R = 180 - 110 = 70 degrees
Verification: 70 + 110 + 70 + 110 = 360 degrees. Correct.
Answer: angle R = 70 degrees, angle S = 110 degrees.
Example 6: Example 6: Isosceles trapezium
Problem: In an isosceles trapezium ABCD with AB || CD, angle A = 60 degrees. Find all angles.
Solution:
In an isosceles trapezium, the base angles are equal.
- angle A = angle B = 60 degrees (base angles on the longer base)
Co-interior angles are supplementary:
- angle A + angle D = 180, so angle D = 120 degrees
- angle B + angle C = 180, so angle C = 120 degrees
Verification: 60 + 60 + 120 + 120 = 360 degrees. Correct.
Answer: angle A = 60, angle B = 60, angle C = 120, angle D = 120 degrees.
Example 7: Example 7: Perimeter of a trapezium
Problem: A trapezium has parallel sides 15 cm and 9 cm. The two legs are 5 cm and 7 cm. Find the perimeter.
Solution:
Given:
- Parallel sides: 15 cm and 9 cm
- Legs: 5 cm and 7 cm
Perimeter:
- Perimeter = 15 + 9 + 5 + 7 = 36 cm
Answer: The perimeter is 36 cm.
Example 8: Example 8: Right trapezium
Problem: A right trapezium has parallel sides 20 cm and 12 cm, with the perpendicular leg being 6 cm. Find the area.
Solution:
Given:
- a = 20 cm, b = 12 cm
- Perpendicular leg = 6 cm = height (h)
In a right trapezium, the perpendicular leg IS the height.
- Area = (1/2) x (20 + 12) x 6
- Area = (1/2) x 32 x 6
- Area = 96 cm²
Answer: The area is 96 cm².
Example 9: Example 9: Field in trapezium shape
Problem: A field in the shape of a trapezium has parallel sides 100 m and 70 m. The distance between the parallel sides is 40 m. Find the area in hectares.
Solution:
Given:
- a = 100 m, b = 70 m, h = 40 m
Area in m²:
- Area = (1/2) x (100 + 70) x 40
- Area = (1/2) x 170 x 40
- Area = 3400 m²
Converting to hectares:
- 1 hectare = 10,000 m²
- Area = 3400/10000 = 0.34 hectares
Answer: The area is 3400 m² or 0.34 hectares.
Example 10: Example 10: Finding diagonal of isosceles trapezium
Problem: An isosceles trapezium has parallel sides 16 cm and 10 cm, and each leg is 5 cm. Find the height.
Solution:
Drop perpendiculars from the ends of the shorter base to the longer base.
The extra length on each side = (16 - 10) / 2 = 3 cm.
Each perpendicular, the leg, and the 3 cm form a right triangle.
- leg² = height² + 3²
- 5² = height² + 9
- 25 = height² + 9
- height² = 16
- height = 4 cm
Answer: The height is 4 cm.
Real-World Applications
Trapezium shapes appear in many real-world objects and situations:
- Dams and Canals: The cross-section of most dams and irrigation canals is trapezoidal. Engineers use the trapezium area formula to calculate water flow capacity.
- Architecture: Trapezoidal windows, roof sections, and facade panels are common in modern buildings.
- Bridges: Some bridge supports and girders have trapezoidal cross-sections for structural strength.
- Table Tops: Trapezoidal tables are used in classrooms to arrange students in a circular pattern.
- Bucket/Tumbler: The cross-section of a bucket or glass tumbler (wider at top, narrower at bottom) is a trapezium.
- Land Measurement: Irregularly shaped plots of land are often divided into trapeziums for area calculation.
- Speed-Time Graphs: In physics, the area under a trapezoidal speed-time graph gives the distance travelled during acceleration.
Key Points to Remember
- A trapezium has exactly one pair of parallel sides (called bases).
- The non-parallel sides are called legs.
- Area = (1/2) x (sum of parallel sides) x height.
- Co-interior angles (same side) are supplementary (sum = 180 degrees).
- An isosceles trapezium has equal legs, equal base angles, and equal diagonals.
- A right trapezium has one leg perpendicular to the bases.
- The diagonals of a trapezium do NOT bisect each other (unlike a parallelogram).
- A trapezium is NOT a parallelogram.
- The sum of all angles of a trapezium is 360 degrees.
- An isosceles trapezium has 1 line of symmetry.
Practice Problems
- Find the area of a trapezium with parallel sides 14 cm and 10 cm, and height 6 cm.
- A trapezium has area 168 cm², parallel sides 16 cm and 12 cm. Find the height.
- One parallel side of a trapezium is 22 cm, height is 10 cm, and area is 200 cm². Find the other parallel side.
- In trapezium ABCD with AB || CD, angle B = 75 degrees. Find angle C.
- An isosceles trapezium has parallel sides 20 cm and 12 cm, and each leg is 5 cm. Find its area.
- Find the perimeter of a trapezium with sides 13 cm, 8 cm, 10 cm, and 7 cm.
- A trapezoidal field has parallel sides 120 m and 80 m with height 50 m. Find the area in hectares.
- In an isosceles trapezium, one base angle is 55 degrees. Find all four angles.
Frequently Asked Questions
Q1. What is a trapezium?
A trapezium is a quadrilateral with exactly one pair of parallel sides. The parallel sides are called bases and the non-parallel sides are called legs.
Q2. Is a trapezium a parallelogram?
No. A parallelogram has BOTH pairs of opposite sides parallel. A trapezium has only ONE pair of parallel sides.
Q3. What is an isosceles trapezium?
An isosceles trapezium is a trapezium in which the two non-parallel sides (legs) are equal in length. Its base angles are equal and its diagonals are equal.
Q4. How do you find the area of a trapezium?
Area = (1/2) x (sum of parallel sides) x height = (1/2) x (a + b) x h.
Q5. What is a right trapezium?
A right trapezium has one leg perpendicular to both parallel sides. Two of its angles are 90 degrees.
Q6. Do the diagonals of a trapezium bisect each other?
No. The diagonals of a trapezium do NOT bisect each other. This property belongs to parallelograms.
Q7. What is the sum of angles in a trapezium?
The sum of all four interior angles of a trapezium is 360 degrees, like any quadrilateral.
Q8. What is the difference between trapezium and trapezoid?
In British English (used in India), a trapezium has one pair of parallel sides. In American English, the same shape is called a trapezoid.
Q9. Are the diagonals of an isosceles trapezium equal?
Yes. The diagonals of an isosceles trapezium are equal in length. This is a distinguishing property of the isosceles trapezium.
Q10. What are co-interior angles in a trapezium?
Co-interior angles are the pair of angles on the same side between the parallel lines. In a trapezium, co-interior angles are supplementary — they add up to 180 degrees.
