Question
Find the area of an irregular shape drawn on a 1 cm grid paper. The shape covers 15 full squares, 8 more-than-half squares, 4 less-than-half squares, and 6 exactly-half squares. What is the estimated area?
(CBSE Class 6 & 7 — mensuration)
Solution — Step by Step
On grid/graph paper, use these rules for each square:
- Full squares: Count as 1
- More than half covered: Count as 1
- Exactly half covered: Count as
- Less than half covered: Count as 0 (ignore)
This gives an estimate, not an exact area. The finer the grid (smaller squares), the better the estimate. This is actually the basic idea behind integration — making the grid infinitely fine.
Why This Works
The grid method works by approximating the irregular boundary with square units. We overcount some regions and undercount others, but the errors roughly cancel out when we use the half-square rule.
graph TD
A["Area of Irregular Shape"] --> B{"Available info?"}
B -->|"Shape on grid paper"| C["Grid Counting Method<br/>Count full + half squares"]
B -->|"Can split into<br/>regular shapes"| D["Decomposition Method<br/>Sum of regular areas"]
B -->|"Equation of boundary<br/>(higher classes)"| E["Integration<br/>∫ y dx"]
C --> F["Quick estimate<br/>Good for Class 6-7"]
D --> G["Exact answer<br/>Split into triangles,<br/>rectangles, trapeziums"]
E --> H["Exact answer<br/>For curves (Class 12)"]Alternative Method — Decomposition
If the irregular shape can be broken into known shapes (rectangles, triangles, semicircles), calculate each area separately and add:
This gives an exact answer when the decomposition is clean.
For CBSE boards: when drawing the shape on graph paper, use a sharp pencil and mark each square clearly as full, half, or less-than-half. The examiner checks your counting table, not just the final answer. Show the table: Full = 15, More than half = 8, Half = 6, Less than half = 4.
Common Mistake
Students count “more than half” squares as 1/2 instead of 1, or they include “less than half” squares in the count. The rule is generous with more-than-half (count as full) and strict with less-than-half (count as zero). This balances out the overestimates and underestimates. Getting this convention wrong can change your answer by 30-40%.