Trapezium

Properties

• One pair of opposite sides is parallel.
• The angles at the two ends of each slanting side add up to 180°.
• If the slanting sides are equal, it is an isosceles trapezium and its diagonals are equal.
• The perimeter is the sum of all four sides.

Why the formulas work

Take two copies of the trapezium, turn one upside down and join them. Together they make a parallelogram with base a + b and the same height. Its area is (a + b) × h, and the trapezium is half of that. Another way to see it: the area is the average of the parallel sides times the height.

Worked example

Take a trapezium with parallel side a 8 cm, parallel side b 14 cm and height 6 cm.

1. Area. Area = ½ (a + b) × h. Here: ½ × (8 + 14) × 6 = 66 cm².

Finding a missing measurement

Height = 2 × area ÷ (a + b). A trapezium of area 66 cm² with parallel sides of 8 cm and 14 cm has a height of 132 ÷ 22 = 6 cm. A common exam form gives the parallel sides as a ratio: sides in the ratio 3 : 5, a height of 6 cm and an area of 96 cm² mean that the sides add up to 2 × 96 ÷ 6 = 32 cm, so they are 12 cm and 20 cm.

Mistakes to avoid

• Adding all four sides into the area formula. Only the two parallel sides are used.
• Using a slanting side as the height.
• Forgetting the half.

Where you see it

The side view of a bucket, a handbag, a lampshade, a bridge support and many plots of land between two parallel roads are trapeziums.

Try these

Work each one out, then tap to check.

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