9. Trapezium Formulas
A trapezium is a quadrilateral having one pair of opposite sides parallel.
The two parallel sides are called the parallel sides or bases. The perpendicular distance between them is called the height.
In this section, we will learn the important formulas for area, perimeter, height, median and special cases.
What Is a Trapezium?
a, b = parallel sides | h = perpendicular height | c, d = non-parallel sides
The height of a trapezium is the perpendicular distance between the two parallel sides. It is not necessarily equal to either of the non-parallel sides.
1. Area of a Trapezium
The most important formula for a trapezium is its area formula.
If the parallel sides are a and b, and the height is h:
Example 1: Find the Area
A trapezium has parallel sides of 12 cm and 20 cm. Its height is 8 cm. Find its area.
= ½(12 + 20) × 8
= ½ × 32 × 8
= 16 × 8
A = 128 cm²
Answer: 128 cm²
Example 2: Find the Area with Different Units
The parallel sides of a trapezium are 15 m and 25 m. Its height is 10 m. Find the area.
= ½ × 40 × 10
A = 200 m²
Answer: 200 m²
2. Height of a Trapezium
Starting from:
we can rearrange the formula to find the height:
Example
The area of a trapezium is 180 cm². Its parallel sides are 10 cm and 20 cm. Find its height.
= (2 × 180) ÷ (10 + 20)
= 360 ÷ 30
h = 12 cm
Answer: 12 cm
3. Finding a Missing Parallel Side
The area formula can also be rearranged to find one of the parallel sides.
Therefore, if b is known:
Example
The area of a trapezium is 150 cm². The height is 10 cm and one parallel side is 12 cm. Find the other parallel side.
= (2 × 150 ÷ 10) − 12
= 30 − 12
a = 18 cm
Answer: 18 cm
4. Perimeter of a Trapezium
The perimeter is the sum of all four sides.
If the four sides are a, b, c and d:
Example
The four sides of a trapezium are 8 cm, 12 cm, 10 cm and 14 cm. Find its perimeter.
= 8 + 12 + 10 + 14
P = 44 cm
Answer: 44 cm
5. Median of a Trapezium
The line segment joining the midpoints of the two non-parallel sides is called the median or midline of a trapezium.
Its length is equal to half the sum of the parallel sides.
If the median is represented by m:
Example
The parallel sides of a trapezium are 14 cm and 22 cm. Find the length of its median.
= ½(14 + 22)
= ½ × 36
m = 18 cm
Answer: 18 cm
6. Area Using the Median
Since:
the area formula can also be written as:
or:
Example
The median of a trapezium is 15 cm and its height is 8 cm. Find its area.
= 15 × 8
A = 120 cm²
Answer: 120 cm²
7. Isosceles Trapezium
An isosceles trapezium has equal non-parallel sides.
If the parallel sides are a and b, and each non-parallel side is c, then the height can be found by forming right-angled triangles.
This formula applies when b > a.
Example
An isosceles trapezium has parallel sides 10 cm and 20 cm. Each non-parallel side is 13 cm. Find its height.
= √[13² − ((20 − 10)/2)²]
= √[169 − 5²]
= √(169 − 25)
= √144
h = 12 cm
Answer: 12 cm
8. Right Trapezium
A right trapezium has at least one pair of adjacent right angles.
In a right trapezium, one of the non-parallel sides may be equal to the height.
The area is still calculated using:
9. Special Case: When Both Parallel Sides Are Equal
If the two parallel sides become equal, the figure becomes a parallelogram.
Suppose:
Then:
which is the usual area formula for a parallelogram.
10. Combined Exam Example
A trapezium has parallel sides of 16 cm and 24 cm, height 10 cm, and the other two sides are 12 cm and 14 cm. Find:
- Area
- Perimeter
- Median
Step 1: Area
= ½(16 + 24) × 10
= ½ × 40 × 10
A = 200 cm²
Step 2: Perimeter
= 16 + 24 + 12 + 14
P = 66 cm
Step 3: Median
= ½(16 + 24)
m = 20 cm
Final Answers:
- Area = 200 cm²
- Perimeter = 66 cm
- Median = 20 cm
11. Common Mistakes in Trapezium Questions
Mistake 1: Forgetting the ½
The area formula is:
Mistake 2: Using the wrong height
The height must be the perpendicular distance between the parallel sides.
Mistake 3: Confusing perimeter and area
Perimeter:
Area:
Mistake 4: Confusing median with a side
The median is:
It is the average of the two parallel sides.
12. Trapezium Formula Chart
| Concept | Formula |
|---|---|
| Area | ½(a + b)h |
| Perimeter | a + b + c + d |
| Height | 2A ÷ (a + b) |
| Median / Midline | ½(a + b) |
| Area Using Median | mh |
| Missing Parallel Side | (2A ÷ h) − known parallel side |
| Isosceles Trapezium Height | √[c² − ((b − a)/2)²] |
13. Practice Questions
Question 1: The parallel sides of a trapezium are 12 cm and 18 cm and its height is 10 cm. Find its area.
Question 2: The four sides of a trapezium are 8 cm, 12 cm, 15 cm and 20 cm. Find its perimeter.
Question 3: The area of a trapezium is 240 cm². Its parallel sides are 12 cm and 28 cm. Find its height.
Question 4: The parallel sides of a trapezium are 16 cm and 24 cm. Find its median.
Question 5: The median of a trapezium is 18 cm and its height is 12 cm. Find its area.
Question 6: An isosceles trapezium has parallel sides 10 cm and 26 cm and each non-parallel side is 17 cm. Find its height.
Question 7: A trapezium has area 300 cm², height 12 cm and one parallel side is 18 cm. Find the other parallel side.
14. Trapezium Formula Quick Revision
15. Mensuration Exam Tip
Whenever you see a trapezium in an exam, identify the two parallel sides first.
Then identify the perpendicular height.
For area questions, immediately think:
For perimeter questions, simply add all four sides.
In Part 10 — Circle Formulas, we will cover radius, diameter, circumference, area, arc-related basics and important circle formulas with diagrams, explanations, solved examples, practice questions and exam tips.