Area is measured in square units and volume in cubic units. Half the marks lost in this topic are lost by using the wrong formula, and the other half by mixing up centimetres and metres.
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Six formulas cover every plane figure in the paper.
⬜ Area formulas
| Shape | Area | Note |
|---|---|---|
| Rectangle | l × b | perimeter 2(l + b) |
| Square | a × a | perimeter 4a |
| Triangle | ½ × base × height | height is the perpendicular one |
| Parallelogram | base × height | not base × slanting side |
| Trapezium | ½ × (a + b) × h | a and b are the parallel sides |
| Rhombus | ½ × d₁ × d₂ | the two diagonals |
| Circle | π r² | circumference 2πr |
The height of a parallelogram is the perpendicular distance, not the slanting side. If a question gives you both 5 cm and 6 cm for a parallelogram, read which one is marked as the height.
Take π as 22/7 when the radius is a multiple of 7. For r = 7, the area is 22/7 × 49 = 154 and the circumference is 2 × 22/7 × 7 = 44. Clean numbers are a hint that you are on the right path.
Volume fills the shape; surface area wraps it.
📦 Solid formulas
| Solid | Volume | Surface area |
|---|---|---|
| Cuboid | l × b × h | 2(lb + bh + hl) |
| Cube | a³ | 6a² |
| Cylinder | π r² h | curved 2πrh; total 2πr(r + h) |
Which surface area does the question want? A closed tin needs the total surface area. An open pipe or a label round a tin needs only the curved surface area. Read the words "open", "closed" and "label" carefully.
Units. 1 m = 100 cm, so 1 m² = 10 000 cm² and 1 m³ = 1 000 000 cm³. Also 1 litre = 1000 cm³, which is how a tank question turns volume into capacity.
Three question types that come round every year.
Paths and borders. A 2 m path runs round a 20 m × 15 m garden. Work out the big area (24 × 19 = 456) minus the small (20 × 15 = 300); the path is 156 m². Remember the path adds twice its width to each dimension.
Working backwards. "A square has area 144 cm². Find its perimeter." The side is √144 = 12, so the perimeter is 48 cm.
Doubling. Double the side of a square and its area becomes four times as big. Double the edge of a cube and its volume becomes eight times as big. Papers love this one.
12 Olympiad questions on area, surface area and volume. Take π as 22/7. 🍀
