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Mixed area problems with parallelograms, triangles and trapeziums
Discover Mathematics P6: Term 2, Unit 12, Lessons 3–4
GoalChoose the right area formula and solve problems with mixed shapes.
New words
compound shape · a shape made of two or more simple shapesformula · a rule written with letters or numberssquare units · cm², m²: the units we use for area
Explanation
To solve a mixed area problem, first name the shape, then pick its formula. Parallelogram: base × height. Triangle: base × height ÷ 2. Trapezium: (a + b) × height ÷ 2. For a compound shape, cut it into simple shapes, find each area, and add them (or subtract a part that is missing). Check that every length is in the same unit before you multiply, and write the answer in square units. A quick check: a triangle with the same base and height as a rectangle must have half its area.
| Shape | Area |
|---|---|
| Parallelogram | b × h |
| Triangle | b × h ÷ 2 |
| Trapezium | (a + b) × h ÷ 2 |
Worked examples
A plot is a rectangle 12 m by 6 m with a triangle on one side. The triangle has base 12 m and height 4 m. Rectangle: 12 × 6 = 72 m². Triangle: 12 × 4 ÷ 2 = 24 m². Total 72 + 24 = 96 m².
A window is a trapezium with parallel sides 80 cm and 120 cm and height 50 cm. Area = (80 + 120) × 50 ÷ 2 = 200 × 50 ÷ 2 = 5,000 cm².
Which is bigger: a parallelogram with base 9 cm and height 5 cm, or a trapezium with parallel sides 6 cm and 8 cm and height 7 cm? Parallelogram: 45 cm². Trapezium: (6 + 8) × 7 ÷ 2 = 49 cm². The trapezium is bigger.
Class activity
Floor plan. Draw the floor of a room made of a rectangle and a triangle on squared paper. Choose lengths in metres. Find the area of the whole floor and the cost of tiles at 80 LE for each square metre.
Practice
1
Find the area of a trapezium with parallel sides 5 cm and 9 cm and height 6 cm.
(5 + 9) × 6 ÷ 2 = 42 cm².
2
A triangle has base 9 cm and height 8 cm. Find its area.
9 × 8 ÷ 2 = 36 cm².
3
A shape is a rectangle 10 cm by 5 cm with a triangle on top (base 10 cm, height 3 cm). Find the total area.
Rectangle 50 cm². Triangle 10 × 3 ÷ 2 = 15 cm². Total 65 cm².
4
A tile is a parallelogram with base 20 cm and height 15 cm. How many such tiles cover 3,000 cm²?
One tile: 20 × 15 = 300 cm². 3,000 ÷ 300 = 10 tiles.
5
A flag is a trapezium with parallel sides 30 cm and 50 cm and height 20 cm. Is its area more than 700 cm²?
(30 + 50) × 20 ÷ 2 = 800 cm². Yes, it is more than 700 cm².