Mathematics Year 7: Areas of the trapezium, the rhombus and the kite
Mathematics, Year 7 (Portugal): Derive the area formulas for the trapezium, rhombus and kite from triangles and parallelograms you already know.
Three formulas built from shapes you already know
A trapezium has one pair of parallel sides, the bases (the longer B and the shorter b); its height h is the distance between them, measured at right angles. Area = (B + b) × h ÷ 2. A kite and a rhombus have two diagonals that cross at right angles, of lengths D and d. Area = D × d ÷ 2. A rhombus is a special kite with four equal sides, so the same formula works for both. Each formula is built from triangles or parallelograms, whose area you can already find.
Why the formulas work
Trapezium: a diagonal cuts it into two triangles with the same height h, one with base B and one with base b. Area = B × h ÷ 2 + b × h ÷ 2 = (B + b) × h ÷ 2. Or: two equal trapeziums, one turned half a turn, make a parallelogram of base B + b and height h, and each trapezium is half of it. Kite: the diagonal D that is the axis of symmetry cuts it into two equal triangles, each with base D and height d ÷ 2, where d is the other diagonal. Area = 2 × D × (d ÷ 2) ÷ 2 = D × d ÷ 2. A rhombus is a kite, so it works too.
Calculating all three areas with numbers
A trapezium has bases 10 cm and 6 cm, height 4 cm and a slanted side of 5 cm: A = (10 + 6) × 4 ÷ 2 = 32 cm². Check with triangles: 10 × 4 ÷ 2 = 20 and 6 × 4 ÷ 2 = 12, and 20 + 12 = 32. A rhombus with diagonals 6 cm and 8 cm: A = 6 × 8 ÷ 2 = 24 cm². Since it is also a parallelogram with side 5 cm, its height is 24 ÷ 5 = 4.8 cm. A kite (the toy) whose two sticks measure 90 cm and 60 cm: A = 90 × 60 ÷ 2 = 2700 cm², which is 0.27 m².
The trap: using the measurements you can see
In the trapezium, using the slanted side (5 cm) instead of the height (4 cm) gives (10 + 6) × 5 ÷ 2 = 40 cm², but the area is 32 cm²: the height is measured at right angles. In the rhombus, 6 × 8 = 48 is double the true area, because the ÷ 2 is missing (48 cm² is the area of the rectangle around it). And 5 × 5 = 25 fails too: the rhombus is leaning over, its height is 4.8 cm, not 5 cm.
Where you see this outside school
A plot shaped like a trapezium, with parallel sides of 30 m and 20 m and 12 m between them, has (30 + 20) × 12 ÷ 2 = 300 m². A trapezium-shaped table top with bases 1.2 m and 0.8 m and a height of 0.5 m has (1.2 + 0.8) × 0.5 ÷ 2 = 0.5 m². For a rhombus-shaped piece, such as a tile or a stained-glass pane, measuring the two diagonals is enough to know how much material it needed.
Test what you learned
Why is the area of a trapezium with bases B and b and height h equal to (B + b) × h ÷ 2?
- a) Because it is half of a rectangle with sides B and b
- b) Two equal trapeziums, one turned half a turn, make a parallelogram of base B + b and height h
- c) Because you use the longer base as in a triangle: B × h ÷ 2
Correct answer: b) Two equal trapeziums, one turned half a turn, make a parallelogram of base B + b and height h — Right: the parallelogram has area (B + b) × h and the trapezium is half of it. A diagonal cutting it into two triangles gives the same result.
A kite has diagonals of 60 cm and 40 cm that cross at right angles. What is its area?
- a) 1200 cm²
- b) 2400 cm²
- c) 100 cm²
- d) 50 cm²
Correct answer: a) 1200 cm² — Right: 60 × 40 ÷ 2 = 1200 cm².
A trapezium has bases of 10 cm and 6 cm, a height of 4 cm and a slanted side of 5 cm. Ana wrote (10 + 6) × 5 ÷ 2 = 40 cm². What is the correct area?
- a) 40 cm²
- b) 64 cm²
- c) 32 cm²
Correct answer: c) 32 cm² — Right: (10 + 6) × 4 ÷ 2 = 32 cm². The slanted side does not enter the calculation.
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