Week 27 · Area of any triangle and of a trapezium · 3 lessons
2
Area of a trapezium by cutting and joining
Discover Mathematics P6: Term 2, Unit 12, Lessons 3–4
GoalFind the area of a trapezium using two copies or two triangles.
New words
trapezium · a four-sided shape with one pair of parallel sidesparallel sides · the two sides of a trapezium that never meetheight of a trapezium · the perpendicular distance between the two parallel sides
Explanation
Take two copies of the same trapezium and turn one upside down. Put them together and they make a parallelogram. Its base is the sum of the two parallel sides, (a + b), and its height is the height of the trapezium, h. The parallelogram has area (a + b) × h, and the trapezium is half of it. So the area of a trapezium is (a + b) × h ÷ 2. Always add only the PARALLEL sides. The two slanted sides are not used. Another way: a diagonal cuts the trapezium into two triangles with the same height h, one with base a and one with base b. Their areas are a × h ÷ 2 and b × h ÷ 2, and together they give the same result.
ƒFormula
Area of a trapezium = (a + b) × h ÷ 2 (a and b are the parallel sides)
Worked examples
The parallel sides are 6 cm and 10 cm and the height is 4 cm. Area = (6 + 10) × 4 ÷ 2 = 16 × 4 ÷ 2 = 32 cm².
Same trapezium, cut by a diagonal. The triangle on the 10 cm side has area 10 × 4 ÷ 2 = 20 cm². The triangle on the 6 cm side has area 6 × 4 ÷ 2 = 12 cm². Total 20 + 12 = 32 cm².
A table top is a trapezium with parallel sides 7 m and 11 m and height 5 m. Area = (7 + 11) × 5 ÷ 2 = 18 × 5 ÷ 2 = 45 m².
Class activity
Two copies. Draw a trapezium on paper and make two copies. Cut them out, turn one around and join them to make a parallelogram. Measure its base and height and check the area formula.
Practice
1
A trapezium has parallel sides 5 cm and 9 cm and height 6 cm. Find its area.
(5 + 9) × 6 ÷ 2 = 14 × 6 ÷ 2 = 42 cm².
2
A trapezium has parallel sides 8 m and 12 m and height 3 m. Find its area.
(8 + 12) × 3 ÷ 2 = 30 m².
3
In the picture, which sides are added, and what is the area?
The parallel sides 10 cm and 6 cm. Area = (10 + 6) × 4 ÷ 2 = 32 cm².
4
A trapezium has area 60 cm² and parallel sides 8 cm and 12 cm. Find the height.
(8 + 12) = 20. 60 × 2 ÷ 20 = 6 cm.
5
Seif finds the area of a trapezium with parallel sides 4 and 8 and height 5 as (4 + 8) × 5 = 60. What did he forget?
To divide by 2. The area is 60 ÷ 2 = 30 square units.
