What is a trapezium?
A trapezium is a shape with four straight sides where one pair of sides is parallel. The parallel sides are the bases. The other two sides are the legs. The straight distance between the two bases is the height.
In the United States this shape is called a trapezoid. In British and Indian books, trapezium is the usual name.
Angles along one leg always add up to 180°. For example, if the angle at the bottom of a leg is 70°, the angle at the top of the same leg is 110°. This is because the bases are parallel, so the angles on the same side of a leg are co-interior angles.
Isosceles trapezium
In an isosceles trapezium the two legs are equal. This gives several facts:
- The two angles on the same base are equal.
- The two diagonals are equal in length.
- The shape is a mirror image across the line that joins the midpoints of the two bases.
Each top angle is 180° minus the base angle. For base angles of 65°, the top angles are 115°.
Right trapezium
In a right trapezium one leg is perpendicular to the bases. That leg is also the height. Two of the corners are 90°. The other leg is slanted, and you can find its length with the Pythagoras theorem: the slanted leg is the longest side of a right triangle whose legs are the height and the difference of the bases.
Midline of a trapezium
The midline (also called the median) joins the midpoints of the two legs. It has two key properties:
- It is parallel to both bases.
- Its length is half the sum of the bases: m = (a + b) ÷ 2.
So the midline is exactly the average of the two bases. If the bases are 8 and 4, the midline is 6. The midline also tells you the area: area = midline × height.
Try it
Cut a trapezium from paper. Fold it so the two ends of one leg meet and crease to mark its midpoint. Do the same for the other leg and draw the midline. Measure the bases and the midline with a ruler. Is the midline exactly the average of the two bases?
Key formulas and definitions
- Midline m = (a + b) ÷ 2
- Area = ½ × (a + b) × h = m × h
- Angles on the same leg: A + D = 180°
- Isosceles trapezium: base angles equal, legs equal, diagonals equal
- Slanted leg of a right trapezium = √(h² + (a − b)²)
Worked examples
1. The bases of a trapezium are 8 cm and 4 cm. Find the midline.
m = (8 + 4) ÷ 2 = 6 cm.
2. The midline of a trapezium is 9 cm and one base is 12 cm. Find the other base.
a + b = 2 × 9 = 18, so the other base = 18 − 12 = 6 cm.
3. In a trapezium the angle at the bottom of one leg is 70°. Find the angle at the top of the same leg.
The two angles on one leg add up to 180°, so the angle is 180° − 70° = 110°.
4. An isosceles trapezium has base angles of 65°. Find all four angles.
Two base angles are 65°. The top angles are 180° − 65° = 115° each. Check: 65 + 65 + 115 + 115 = 360°.
5. A right trapezium has bases 9 cm and 5 cm and height 3 cm. Find the slanted leg and the perimeter.
Difference of bases = 4 cm. Slanted leg = √(3² + 4²) = 5 cm. Perimeter = 9 + 5 + 3 + 5 = 22 cm.
6. The midline of a trapezium is 14 cm. The bases are in the ratio 3 : 4. Find the bases.
a + b = 28. Let the bases be 3k and 4k, so 7k = 28 and k = 4. The bases are 12 cm and 16 cm.
Common mistakes
- Saying the midline equals one base. It is the average of both bases.
- Forgetting to divide by 2 when finding the midline: (a + b) ÷ 2, not a + b.
- Thinking both pairs of sides must be parallel. That would be a parallelogram. A trapezium needs only one pair.
- Using the slanted leg as the height. The height is the straight distance between the bases, at 90°.