Mensuration

Mensuration is a formula topic, but the marks are lost on units and on reading the wrong surface, not on the formulas.

1. Plane figures. rectangle l × b, square a², triangle ½ × base × height, parallelogram base × height, trapezium ½(a + b) × h, rhombus ½ d₁ d₂, circle πr² with circumference 2πr. For a triangle with three sides known, use Heron's rule with s = (a+b+c)/2.
2. Solids. cuboid l b h with total surface 2(lb + bh + hl); cube a³ with surface 6a²; cylinder πr²h, curved surface 2πrh, total 2πr(r + h); cone ⅓πr²h, curved surface πrl where the slant l = √(r² + h²); sphere ⁴⁄₃πr³ with surface 4πr².
3. Scaling. If every length is multiplied by k, area is multiplied by k² and volume by k³. So a side increased by x% raises the area by 2x + x²/100 percent.
4. Units. Convert everything into one unit before the first multiplication: 1 m = 100 cm, 1 m² = 10,000 cm², 1 m³ = 10,00,000 cm³, 1 hectare = 10,000 m², and 1 litre = 1,000 cm³.

The single most common mistake is mixing units inside one calculation — multiplying a length in metres by a breadth in centimetres. It never announces itself in the answer; it merely gives a number that looks plausible. Fix the unit before you start, not afterwards.

A close second is answering with the curved surface area when the total was asked, or the reverse. Read the words "open at the top", "closed", "curved" and "total" carefully: an open cylindrical tank has 2πrh + πr², with only one circle, not two.

triangle ½bh · trapezium ½(a+b)h · rhombus ½d₁d₂ · circle πr², 2πr · cuboid lbh, 2(lb+bh+hl) · cylinder πr²h, curved 2πrh, total 2πr(r+h) · cone ⅓πr²h, πrl with l = √(r²+h²) · sphere ⁴⁄₃πr³, 4πr² · length × k → area × k², volume × k³ · 1 hectare = 10,000 m², 1 litre = 1,000 cm³

The side of a square field is increased by 20%. By what percent does its area increase? → Every length is multiplied by 1.20, so the area is multiplied by 1.20² = 1.44, an increase of 44%. Sanity check two ways: the formula 2x + x²/100 gives 40 + 4 = 44; and with real numbers, a side of 10 m has an area of 100 m², while a side of 12 m has 144 m², which is 44 m² more on a base of 100. Note that the answer is not 20% and not 40%.