Area of a trapezium

The idea

Area of a trapezium = 1/2 × height × sum of the parallel sides = 1/2 h(a + b), found by splitting it into a rectangle and two triangles, or as half the parallelogram formed by joining two copies of it.

A four-sided field

A trapezium has two parallel sides, 4 and 9 long, and the distance between them is 4. We know the area of a rectangle, a triangle and a parallelogram.

How can we find the area of the trapezium?

• Multiply the two parallel sides and the height
• Join it with a copy of itself to make a parallelogram, and take half
• Add up the four sides

Two copies

Here are two copies of the trapezium. Press Turn the copy for a half turn, then move it so that its slanted side meets the slanted side of the first one. Then open the second tab and tap the three pieces.

Half of a parallelogram

Area of a trapezium = ½ × height × sum of the parallel sides = ½ h (a + b), found by splitting it into a rectangle and two triangles, or as half the parallelogram formed by joining two copies of it.

Two copies, one turned half a turn, fit together along a slanted side. The angles next to a slanted side add up to 180°, so the sides line up straight and the joined figure is a parallelogram. Its base is a + b = 4 + 9 = 13 and its height is the same 4, so its area is 13 × 4 = 52. The trapezium is half of it: 26 = ½ × 4 × (4 + 9).

The other way: cut the trapezium into a triangle, a rectangle and a triangle. Their areas ½ × 2 × 4 = 4, 4 × 4 = 16 and ½ × 3 × 4 = 6 add up to 26, the same.

Call the height h, the triangle bases x and y, the rectangle a, and the longer parallel side b. Area = ½hx + ha + ½hy = ½h(x + y + 2a) The long side is b = x + y + a, so x + y = b − a and Area = ½h(b − a + 2a) = ½h(a + b).

Notes

Area of a trapezium = ½ × height × sum of the parallel sides = ½ h (a + b), found by splitting it into a rectangle and two triangles, or as half the parallelogram formed by joining two copies of it.

Check yourself

The parallel sides of a trapezium are 7 cm and 13 cm and the distance between them is 6 cm. What is its area, in cm²?

Answer: 60

Area = ½ × 6 × (7 + 13) = 3 × 20 = 60 cm².

A trapezium has an area of 72 cm² and a height of 6 cm. One parallel side is 8 cm. How long is the other parallel side, in cm?

Answer: 16

72 = ½ × 6 × (8 + b) = 3 × (8 + b), so 8 + b = 24 and b = 16 cm.

Which gives the area of a trapezium with parallel sides a and b and height h?

Each square of the grid is 1 m by 1 m. What is the area of the trapezium field ABCD, in m²?

Answer: 30

The parallel sides are 5 m and 10 m, the height is 4 m. Area = ½ × 4 × (5 + 10) = 30 m².

• (a + b) × h. That is the area of the parallelogram made from two copies of the trapezium. The trapezium is half of it.
• ½ × a × b × h. The parallel sides are added, not multiplied: a + b, because they make the base of the joined parallelogram.
• ½ × h × (a + b) — correct. Yes! Two copies make a parallelogram of base a + b and height h, and the trapezium is half of it.
• h × (a − b). The two parallel sides are added, not subtracted.