Mensuration

Keep two mental shelves and never mix them: PLANE figures give area (square units) and perimeter (linear units); SOLIDS give volume (cubic units) and surface area (square units). Before writing a formula, check the unit the answer must carry — if the question asks for litres of water, you need a volume, so any formula producing square units is wrong. The second habit is scaling: if every length is multiplied by k, area is multiplied by k squared and volume by k cubed. That single rule answers most 'if the radius is doubled' questions without any calculation.

• Area is squared units; volume is cubed units — a formula giving the wrong power is instantly suspect.

  Before trusting a recalled formula, check its dimensional sense: area formulas multiply two lengths (length x length), volume formulas multiply three (length x length x length). A formula that does not match this pattern has been misremembered.

• A cylinder is a stack of circles; a cone is a cylinder squeezed to a point.

  Picturing the solid this way explains why a cylinder's volume is (circle area) x height, and why a cone's volume is exactly one-third of a cylinder with the same base and height — the 'squeeze to a point' loses two-thirds of the volume.

• 1Rectangle: area = length x breadth; perimeter = 2(length + breadth).
• 2Circle: area = pi x r squared; circumference = 2 x pi x r.
• 3Triangle: area = 1/2 x base x height.
• 4Cuboid volume = length x breadth x height; cube volume = side cubed.
• 5Cylinder volume = pi x r squared x height; sphere volume = (4/3) x pi x r cubed; cone volume = (1/3) x pi x r squared x height.

• Circle: area = pi x r squared, circumference = 2 x pi x r. Take pi as 22/7 when the radius is a multiple of 7.
• Cuboid: volume = length x breadth x height, total surface area = 2(lb + bh + hl). Cube of side a: volume = a cubed, surface area = 6 a squared.
• Cylinder: volume = pi x r squared x h, curved surface area = 2 x pi x r x h. Sphere: volume = 4/3 x pi x r cubed, surface area = 4 x pi x r squared.
• Triangle area = 1/2 x base x height; with three sides known, use Heron's formula, area = square root of s(s-a)(s-b)(s-c) where s is half the perimeter.
• Unit conversions that decide the answer: 1 cubic metre = 1000 litres, 1 hectare = 10,000 square metres, and 1 square metre = 10,000 square centimetres.
• If all linear dimensions are multiplied by k, area becomes k squared times and volume k cubed times the original.

• A cone's volume is one-third of a cylinder with the same base radius and height, not one-half — the '1/3' factor is easy to misremember as '1/2'.

• NCERT Class X — Mathematics
• Quantitative Aptitude for Competitive Examinations — R S Aggarwal
• Fast Track Objective Arithmetic — Arihant

• NCERT textbooks (free PDFs)