Quadrilaterals • Topic 3 of 3

Trapezium & Kite

A trapezium (US: trapezoid) has exactly one pair of parallel sides, called the bases a and b, separated by perpendicular height h. Its area is ½ × (a + b) × h — the average of the two parallel sides times the height. In an isosceles trapezium the non-parallel sides are equal, the base angles are equal, and the diagonals are equal. A kite has two distinct pairs of adjacent equal sides. Its diagonals are perpendicular, but only ONE diagonal bisects the other (the axis of symmetry bisects the cross-diagonal). Like the rhombus, a kite’s area is ½ × d₁ × d₂. The link is no accident: a rhombus is simply a kite whose two pairs are all equal, which is why both share the half-product-of-diagonals area rule. In CAT, trapezium area often pairs with dropping a perpendicular to split off a right triangle and recover the height from the slant sides.

✅ Solved examples

1. A trapezium has parallel sides 10 cm and 14 cm with height 6 cm. Find its area.
Area = ½ × (10 + 14) × 6 = ½ × 24 × 6 = 72 cm².
2. A kite has diagonals 18 cm and 8 cm. Find its area.
Area = ½ × d₁ × d₂ = ½ × 18 × 8 = 72 cm².
3. A trapezium has area 80 cm², height 8 cm and one parallel side 6 cm. Find the other parallel side.
80 = ½ × (6 + b) × 8 = 4(6 + b) ⇒ 6 + b = 20 ⇒ b = 14 cm.
4. An isosceles trapezium has parallel sides 8 cm and 20 cm and slant sides 10 cm. Find its area.
Horizontal overhang each side = (20 − 8)/2 = 6. Height = √(10² − 6²) = √64 = 8. Area = ½ × (8 + 20) × 8 = ½ × 28 × 8 = 112 cm².

✏️ Practice — try these, take hints as needed

1. A trapezium has parallel sides 12 cm and 18 cm, height 10 cm. Area?
Area = ½ (a + b) h.
½ × (12 + 18) × 10.
½ × 30 × 10.
150 cm²
2. A kite has diagonals 20 cm and 15 cm. Area?
Area = ½ d₁ d₂.
½ × 20 × 15.
10 × 15.
150 cm²
3. A trapezium has area 126 cm², parallel sides 9 cm and 12 cm. Find the height.
Area = ½ (a + b) h.
126 = ½ × 21 × h.
126 = 10.5 h.
12 cm
4. A kite has area 84 cm² and one diagonal 14 cm. Find the other diagonal.
Area = ½ d₁ d₂.
84 = ½ × 14 × d₂.
84 = 7 d₂.
12 cm
5. An isosceles trapezium has parallel sides 6 cm and 16 cm and slant sides 13 cm. Find its area.
Overhang = (16 − 6)/2 = 5.
Height = √(13² − 5²).
√144 = 12; then ½(6+16)×12.
132 cm²

📝 Topic test — 8 questions

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