270. The Trapezium's Secret · area hides in the parallel sides and height

Try to break it

Drag A and B along the bottom to set base a, C and D along the top to set base b, and the height handle to change h. The area always equals ½ × (a + b) × h. Bring the bases equal (a = b) and the trapezium becomes a parallelogram with area b·h — still consistent with the formula.

How you build it

Construct a trapezium with one pair of parallel sides.

• Point tool: drop A at the bottom-left.
• Point tool: drop B to the right of A — AB is the lower parallel side.
• Segment tool: join A to B — the lower parallel side (a).
• Point tool: drop D well above A — the gap is the height h.
• Parallel tool: click D, then click base AB — the upper parallel side runs exactly parallel to AB through D.
• Point tool: drop C on the upper parallel line — make DC a different length from AB, so it is a trapezium.
• Segment tool: join B to C — the right leg.
• Segment tool: join A to D — the left leg. Area = ½ × (AB + DC) × h.

The proof, step by step

Prove that the area of a trapezium is ½ × (a + b) × height.

• Draw diagonal AC to split the trapezium into two triangles: △ABC and △ADC.
• △ABC has base b and height h. Its area is ½ × b × h.
• △ADC has base a and height h. Its area is ½ × a × h.
• Total Area = Area(△ABC) + Area(△ADC) = ½bh + ½ah = ½(a + b)h. Q.E.D.

Worked example

A trapezium has parallel sides of 10 cm and 14 cm. If its height is 6 cm, what is its area?

Area = ½ × (10 + 14) × 6 = ½ × 24 × 6 = 72 cm².

• 60 cm²
• 72 cm² — correct
• 84 cm²
• 96 cm²