Area of a Trapezium and a Polygon
193. The Trapezium's Secret · Area = ½ × (a + b) × h
The area of a trapezium is always half the sum of parallel sides times the height.
Area of a trapezium = ½ × (sum of parallel sides) × height = ½(a + b)h, where a and b are the lengths of the two parallel sides and h is the perpendicular distance between them.
What this lesson covers
Try to break it
Drag H to change the height, B to change the bottom base, and C to change the top base. The area always equals ½ × (a + b) × h. Try to drag the handles to a configuration where the readout disagrees with the formula — impossible.
How you build it
Construct a trapezium.
- Draw segment AB — the longer parallel side (base) of the trapezium.
- Construct a perpendicular at A — this gives the height direction of the trapezium.
- With centre A, swing an arc on the perpendicular and mark point D — this sets the height h of the trapezium.
- Through D, draw a line parallel to AB — this is the top parallel side of the trapezium.
- Draw segment AD — the left side of the trapezium.
- Draw segment BC to close the trapezium — AB ∥ DC, with height AD.
The proof, step by step
Prove that the area of a trapezium is half the sum of the parallel sides times the height.
- Draw a trapezium ABCD with AB parallel to CD.
- Draw a line through C parallel to DA, meeting AB at E.
- AECD is a parallelogram, so Area(AECD) = CD × h.
- Triangle CEB has base EB = AB - CD and height h.
- Area(Trapezium) = Area(AECD) + Area(CEB) = CD × h + ½ × (AB - CD) × h.
- Simplifying gives ½ × (AB + CD) × h.
Worked example
The parallel sides of a trapezium are 8 cm and 12 cm. If the height is 5 cm, find the area of the trapezium.
Area = ½ × (8 + 12) × 5 = ½ × 20 × 5 = 50 cm².
- 25 cm²
- 50 cm² — correct
- 100 cm²
- 120 cm²