Awkward outlines, ordinary pieces
A notice board is a rectangle with a triangle on top. What is its area?
- The board: 8 across, 5 up, and a triangle 4 high on top.
- The rectangle: 8 × 5 = 40.
- The triangle: half × 8 × 4 = 16.
- Together: 40 + 16 = 56.
Split it into shapes that already have a rule
The area is 56 square units.
There is no rule for the whole board. There are rules for the pieces it is made of. Draw lines between corners, and a many-sided shape splits into triangles. A four-sided shape splits along a diagonal, a corner-to-corner line.
Count every part once, with no gap and no overlap. Subtracting also works. The board fits in an 8 × 9 rectangle, with area 72. The tip is above the middle of the top side. So each empty corner is a triangle, 4 wide and 4 high. Its area is ½ × 4 × 4 = 8. Then 72 − 8 − 8 = 56, the same answer.
A park is 10 m by 6 m. A path 2 m wide runs outside its edge. Each length grows by 2 m at both ends. So 10 + 4 = 14 and 6 + 4 = 10. The outer rectangle is 14 m by 10 m. Its area is 14 × 10 = 140 m². The park is 10 × 6 = 60 m². The path is 140 − 60 = 80 m².
Pick a diagonal. Add the two triangles.
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