Question
A leaf is placed on a grid paper with 1 cm squares. Inside the leaf boundary, there are 15 complete squares, 8 more-than-half squares, and 6 less-than-half squares. Estimate the area of the leaf.
Solution — Step by Step
The standard rule for estimating area using a grid:
- Complete squares: Count fully (each = 1 sq cm)
- More than half squares: Count as 1 sq cm each
- Less than half squares: Ignore them (count as 0)
- Exactly half squares: Count as 0.5 sq cm each
Area
This is an estimate, not an exact area — but it is good enough for practical purposes.
Why This Works
graph TD
A["Finding area of irregular shape"] --> B["Can it be split into regular shapes?"]
B -->|Yes| C["DECOMPOSITION: split into rectangles, triangles, etc."]
B -->|No| D["Is it on a grid?"]
D -->|Yes| E["GRID METHOD: count squares"]
D -->|No| F["Is the boundary defined by a function?"]
F -->|Yes| G["INTEGRATION: area under curve"]
F -->|No| H["Use physical methods: weigh paper cutout"]The grid method works because we are essentially counting unit squares. Each complete square is exactly 1 sq cm. For partial squares, the “more than half = 1, less than half = 0” rule is a rounding approximation that balances out — the overcount of “more than half” squares roughly compensates for the undercount of “less than half” squares.
For more precise results, use a finer grid (smaller squares). This is actually the basic idea behind integration in higher classes — take infinitely small squares and sum them up.
The decomposition method is more accurate when the shape can be broken into standard figures: rectangles, triangles, trapeziums, semicircles. Find the area of each piece and add them up.
Alternative Method
For CBSE exam problems, shapes are often designed to decompose neatly. An L-shaped figure splits into two rectangles. A figure with a semicircular end splits into a rectangle plus a semicircle. Look for straight lines where you can cut the shape.
For a trapezium shape: where and are parallel sides and is the height between them. This formula covers many irregular-looking quadrilaterals.
Common Mistake
Counting less-than-half squares as 1. The grid method only works if you follow the convention consistently. Including less-than-half squares overestimates the area. Similarly, excluding more-than-half squares underestimates it. Stick to the rule: full and more-than-half count as 1; less-than-half count as 0; exactly half counts as 0.5.