A trapezoid is a quadrilateral with exactly one pair of parallel sides (the two bases); the other two sides are the legs. Its area is the average of the two bases times the height: ½ (b₁ + b₂) × h — think of it as a rectangle whose width is the average base. The height is the perpendicular distance between the parallel sides, never a slanted leg. An isosceles trapezoid has equal legs and equal base angles, and because a leg is a transversal across the two parallel bases, the two angles along the same leg are co-interior and therefore supplementary. Running the area formula backward — solving for a base or the height when the area is given — is a standard GRE variation.
✅ Solved examples
1. A trapezoid has parallel sides 8 and 12 and a height of 5. Find its area.
Area = ½ (b₁ + b₂) × h = ½ (8 + 12) × 5 = ½ × 20 × 5 = 50.
2. A trapezoid has an area of 60 and parallel sides of 6 and 14. Find its height.
60 = ½ (6 + 14) × h = ½ × 20 × h = 10h → h = 6.
3. A trapezoid has an area of 45, a height of 5, and one base of 10. Find the other base.
45 = ½ (10 + b) × 5 → (10 + b) × 5 = 90 → 10 + b = 18 → b = 8.
4. In an isosceles trapezoid, a base angle is 65°. Find the adjacent angle along the same leg (at the other base).
The two angles on one leg are co-interior between parallel bases, so they are supplementary: 180° − 65° = 115°.
✏️ Practice — try these, take hints as needed
1. A trapezoid has parallel sides 10 and 6 and a height of 4. Find its area.
Average the bases.
(10 + 6) / 2 = 8.
Area = 8 × 4.
32
2. A trapezoid has an area of 84 and parallel sides of 9 and 12. Find its height.
84 = ½ (9 + 12) × h.
84 = 10.5 × h.
Divide by 10.5.
8
3. A trapezoid has an area of 55, a height of 5, and one base of 7. Find the other base.
55 = ½ (7 + b) × 5.
(7 + b) × 5 = 110.
7 + b = 22.
15
4. In an isosceles trapezoid a base angle is 72°. Find each of the two angles at the shorter base.
Angles along a leg are supplementary.
180 − 72.
Both top angles are equal.
108° each
5. A trapezoid has parallel sides 15 and 9 and a height of 10. Find its area.
Average the bases: (15 + 9)/2.
That is 12.
Area = 12 × 10.
120
📝 Topic test — 8 questions
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