Show how we can use two identical copies of a trapezium to make a parallelogram. How will this give us the formula for the area of a trapezium?
Answer: Turn a copy upside down (half-turn) and place it against a slant side of the first: the bottom becomes b + a and the top a + b, giving a parallelogram with base a + b and height h. Area = (a + b)h, so one trapezium = 12(a + b)h.
Step-by-step solution
Idea: A half-turn copy fits exactly along the shared slant side, and the parallel sides line up end to end.
- Take the trapezium with parallel sides a (top) and b (bottom), height h. Turn an identical copy through 180° and join it along the right slant side.1 mark
- The angles at each joined corner add up to 180° (co-interior angles of the original trapezium), so the bottoms b and a lie on one straight line, and so do the tops a and b. The new shape has both pairs of opposite sides parallel: a parallelogram with base a + b and height h.1 mark
- Area of parallelogram = (a + b)h = 2 × area of trapezium, so area of trapezium = 12(a + b)h.1 mark
Two copies (one turned half round) form a parallelogram of base a + b and height h, so one trapezium has area ½(a + b)h.
Answer to write in the exam
Copy rotated 180°, joined along a slant side
Co-interior angles sum to 180° ⇒ bases in one line ⇒ parallelogram
Base = a + b, height = h ⇒ area = (a + b)h
∴ Area of trapezium = 12(a + b)h
Common mistakes that cost marks
- Placing the copy without turning it; then the sides do not line up.
- Taking the parallelogram’s base as b only.
How this can come in the exam
MCQ (1 mark)
Two copies of a trapezium with parallel sides 6 cm and 10 cm and height 5 cm form a parallelogram. The area of the parallelogram is
- 40 cm2
- 60 cm2
- 160 cm2
- 80 cm2
Show answer
(D) 80 cm2
Base 16, height 5: 80 cm2 (each trapezium 40 cm2).
Try one yourself
Two copies of a trapezium form a parallelogram of base 18 cm and height 9 cm. Find the area of one trapezium.
Show answer
12 × 18 × 9 = 81 cm2.
More questions like this
- Show that the area of a kite is half the product of its diagonals. Show this:
- Three problems about fitting congruent shapes together:
- What fraction of the triangle is shaded? What fraction of the square is shaded?
- What fraction of the rectangle is covered by the circles?
- Use the above to make a conjecture about the area occupied by circles fitted into a rectangle in the manner shown. Test your conjecture for particular cases: 10 circles; 20 circles; 50 circles. Then prove your conjecture!