Plane Figures
| Figure | Perimeter | Area |
|---|---|---|
| Rectangle (l × b) | 2(l + b) | l × b |
| Square (side a) | 4a | a², or diagonal²/2 |
| Triangle | a + b + c | ½ × base × height |
| Equilateral triangle (side a) | 3a | (√3/4)a² |
| Parallelogram | 2(a + b) | base × height |
| Rhombus (diagonals d₁, d₂) | 4a | ½ d₁d₂ |
| Trapezium | Sum of sides | ½ × (sum of parallel sides) × height |
| Circle (radius r) | 2πr | πr² |
| Sector of angle θ | Arc = 2πr × θ/360 | πr² × θ/360 |
Heron's formula
Area = √[s(s − a)(s − b)(s − c)], s = (a + b + c)/2
Sides 13, 14 and 15 cm: s = 21, area = √(21 × 8 × 7 × 6) = √7056 = 84 cm².
- Take π = 22/7 unless told otherwise. A wheel of radius 35 cm covers 2 × 22/7 × 35 = 220 cm in one turn.
- Path around a rectangle l × b of width w (outside): area = 2w(l + b + 2w). Two crossing roads of width w through the middle: w(l + b − w).
- Scaling: if every length is multiplied by k, area is multiplied by k² and volume by k³. Increasing each side of a square by 10% raises its area by 21%.
Solids
| Solid | Curved / lateral surface | Total surface | Volume |
|---|---|---|---|
| Cuboid l × b × h | 2h(l + b) | 2(lb + bh + hl) | lbh |
| Cube (side a) | 4a² | 6a² | a³ |
| Cylinder | 2πrh | 2πr(r + h) | πr²h |
| Cone (slant l = √(r² + h²)) | πrl | πr(r + l) | ⅓πr²h |
| Sphere | 4πr² | 4πr² | ⁴⁄₃πr³ |
| Hemisphere | 2πr² | 3πr² | ⅔πr³ |
- Diagonal of a cuboid = √(l² + b² + h²); of a cube = a√3.
- A cone, a hemisphere and a cylinder with the same radius and height (h = r) have volumes in the ratio 1 : 2 : 3.
- Melting and recasting keeps the volume the same: number of small solids = big volume ÷ small volume.
- 1 m³ = 1000 litres; 1 litre = 1000 cm³.
Common Mistakes
Using the diameter where the radius is needed; using height instead of slant height for a cone's curved surface; forgetting the ⅓ in a cone's volume; and mixing cm and m.