25
Area of a trapezoid
Textbook: pp. 84–86
GoalKnow the formula for the area of a trapezoid and use it in problems.
New words
trapezoid · trapetsiyabases · asoslarmidline · o‘rta chiziqaltitude · balandlik
Explanation
Theorem: the area of a trapezoid equals half the sum of its bases times the altitude: S = (a + b) : 2 · h. Idea of the proof: place two congruent trapezoids together, one turned around against a leg of the other, to get a parallelogram with base a + b and altitude h. Its area is (a + b) · h, and the trapezoid has half of that. Half the sum of the bases is the midline, so S = m · h, where m is the midline. If the diagonals of an isosceles trapezoid are perpendicular, its midline equals its altitude, so its area is S = h².
Worked examples
The bases are 9 cm and 15 cm and the altitude is 7 cm. The area is (9 + 15) : 2 · 7 = 84 cm².
The diagonals of an isosceles trapezoid are perpendicular and its altitude is 10 cm. The area is 10² = 100 cm².
Class activity
“Two trapezoids, one parallelogram”: draw a trapezoid on squared paper, draw a copy turned around next to it, and compute the base of the parallelogram you get.
Practice
1
The bases of a trapezoid are 8 cm and 14 cm and its altitude is 5 cm. What is its area (cm²)?
55
2
A trapezoid has midline 13 cm and altitude 9 cm. What is its area (cm²)?
117
3
A trapezoid has area 90 cm², altitude 6 cm and one base 10 cm. How long is the other base (cm)?
20
4
Why do two congruent trapezoids make a parallelogram, and why is the trapezoid’s area half of the parallelogram’s?
Turning the second one around makes the legs fit, giving base a + b and altitude h; the two congruent halves have equal area, so each is half.