Volume & Surface Area

Picture each solid before you reach for a formula.

• Cuboid — a brick, with length l, breadth b, height h. Volume lbh; total surface 2(lb + bh + hl); diagonal √(l² + b² + h²).
• Cube — a cuboid with all edges equal to a. Volume a³; surface 6a² (six equal square faces).
• Cylinder — a circular tin, radius r, height h. Volume πr²h; curved surface 2πrh; total surface 2πr(r + h) (curved plus the two circular ends).
• Cone — a party hat on the same base, radius r, height h, slant height l = √(r² + h²). Volume ⅓πr²h; curved surface πrl.
• Sphere — a ball of radius r. Volume ⁴⁄₃πr³; surface 4πr².

Two habits make this topic easy:

1. Remember the cone is one third of the cylinder on the same base and height. That one relationship saves memorising a separate cone volume.
2. Melting and recasting preserves volume. Old volume = new volume; nothing else carries over. Note that if the edges are in the ratio k, the volumes are in the ratio k³ — a cube of edge 12 gives (12/3)³ = 64 cubes of edge 3, not 4.

The single most common mistake is confusing volume with surface area. Check the unit in the options: cm³ means volume, cm² means surface area. Take π = 22/7 whenever a radius is a multiple of 7 or is 3.5 — the paper chooses those numbers so the sevens cancel.

cuboid lbh, 2(lb+bh+hl) · cube a³, 6a² · cylinder πr²h, 2πrh · cone ⅓πr²h, πrl · sphere ⁴⁄₃πr³, 4πr² · melting keeps volume

Find the volume of a cylinder of radius 7 cm and height 10 cm, taking π = 22/7. → V = πr²h = 22/7 × 7 × 7 × 10. The 7 in the denominator cancels one 7, leaving 22 × 7 × 10 = 1540 cm³. Sanity check on the units and the neighbours: the curved surface 2πrh = 2 × 22/7 × 7 × 10 = 440 cm² — a different quantity in a different unit; and a cone on the same base and height would hold 1540 ÷ 3 ≈ 513.33 cm³, exactly a third.