Mensuration — length, perimeter, area, volume and surface area

What this heading covers

The measuring of shapes: how long the edge is, how much surface it covers, and how much it holds. The geometry of lines, angles and the shapes themselves has lessons of its own; what is here is what you do once the shape is known.

Weight, time and money have a lesson of their own, and so do patterns and data handling, because the syllabus gives each of them headings of their own.

Almost nothing here needs derivation. It needs the formula exact and the unit right — and the unit is what the paper is actually testing.

Length, and the metric ladder

Each step is a factor of 10:

km  →  hm  →  dam  →  m  →  dm  →  cm  →  mm      (÷10 at each step to the right)

1 km = 1000 m; 1 m = 100 cm; 1 cm = 10 mm.

The prefixes are the same throughout the SI system, which is the point of it: kilo is a thousand, centi a hundredth, milli a thousandth. A question that asks for 0.75 km in centimetres is asking whether you can walk the ladder: 0.75 km = 750 m = 75,000 cm.

Length is measured in metres, area in square metres, volume in cubic metres — one dimension, two, and three. Everything else in this lesson follows from that sentence.

Perimeter, area and volume

Shape	Perimeter / circumference	Area
Square	4 × side	side²
Rectangle	2 × (length + breadth)	length × breadth
Triangle	sum of the three sides	½ × base × height
Equilateral triangle	3 × side	(√3/4) × side²
Circle	2πr	πr²
Semicircle	πr + 2r	½πr²
Parallelogram	2 × (a + b)	base × height
Rhombus	4 × side	½ × d₁ × d₂
Trapezium	sum of the four sides	½ × (sum of parallel sides) × height

When the three sides are known but no height is, use Heron's formula. With s = (a + b + c) ÷ 2,

area = √( s (s − a) (s − b) (s − c) )

A sector of a circle, for an angle θ at the centre: arc length = (θ ÷ 360) × 2πr, and area = (θ ÷ 360) × πr² — in both cases the whole circle scaled by the fraction of the turn.

π is 22/7 or 3.14 to two places. Use 22/7 when the radius is a multiple of 7, which is why the examiner so often chooses 7, 14 or 21; otherwise 3.14 is easier.

The solids

Solid	Volume	Surface area
Cube	side³	6 × side²
Cuboid	l × b × h	2(lb + bh + hl)
Cylinder	πr²h	curved 2πrh; total 2πr(r + h)
Cone	⅓πr²h	curved πrl; total πr(l + r)
Sphere	(4/3)πr³	4πr²
Hemisphere	(2/3)πr³	curved 2πr²; total 3πr²

For a cone the slant height is not the height: l = √(r² + h²), and using h where l is wanted is the standard error in a cone question.

Diagonals: square = side√2; rectangle = √(l² + b²); cuboid = √(l² + b² + h²).

Two relations worth recognising rather than deriving, because a question sometimes states one and asks for the other:

• A cone is one third of the cylinder on the same base and of the same height.
• A sphere is two thirds of the cylinder that just contains it — and the sphere's surface area equals that cylinder's curved surface area.

The units are the question

Perimeter is a length, area is a length squared, volume is a length cubed — so they are measured in cm, cm² and cm³. A multiple-choice question will offer the right number with the wrong unit, and it catches the candidate who stopped one step early.

Three further points the paper sets:

• Two shapes can have the same perimeter and different areas. A 6 × 6 square and a 10 × 2 rectangle both have perimeter 24, and areas of 36 and 20. Of all rectangles with a given perimeter, the square encloses the largest area.
• Two shapes can equally have the same area and different perimeters.
• Doubling every side does not double the area. It multiplies area by 4 and volume by 8, because area goes as the square and volume as the cube. More generally, multiplying every length by k multiplies area by k² and volume by k³.

That last rule is worth holding as a sentence: lengths scale by k, areas by k², volumes by k³. A question about a model, a map or a photograph enlarged is the same rule wearing different clothes.

Converting the units

Length converts by 10s. Area and volume do not.

	Because	So
1 m = 100 cm	—	—
1 m² = 10,000 cm²	100 × 100	not 100
1 m³ = 1,000,000 cm³	100 × 100 × 100	not 100
1 km² = 1,000,000 m²	1000 × 1000	—
1 hectare = 10,000 m²	a 100 m square	1 km² = 100 hectares

Capacity: 1 L = 1000 mL, and 1 litre = 1000 cm³, so 1 mL = 1 cm³. That equivalence is the bridge between a volume question and a capacity one, and it is asked directly: a tank measured in centimetres has its capacity asked for in litres, and the conversion is the whole question.

A cubic metre is 1000 litres, which follows from the same two lines and is the form a water-tank question usually takes.

The problems built on these

Four recurring question types, each of which is a formula with a sentence wrapped round it:

The sentence	What it is really asking
Fencing, railing or putting a border round a field	Perimeter, then × rate per metre
Flooring, tiling, carpeting, whitewashing, papering	Area, then × rate per square metre
A path around or inside a rectangle	Area of the outer rectangle minus the inner one
Painting or papering the four walls of a room	Lateral surface area = 2h(l + b), less doors and windows

For a path of width w running outside an l × b field, the outer rectangle is (l + 2w) × (b + 2w). The 2 is what gets dropped — the path lies along both ends, so each dimension grows by twice its width. A path running inside the field subtracts instead: (l − 2w) × (b − 2w).

Tiles and bricks are the same idea once more: the number needed is the area to be covered divided by the area of one tile, with both in the same unit. A question that gives the room in metres and the tile in centimetres is testing the squared conversion, not the division.

Questions of the kind the paper sets

The perimeter of a rectangle is 36 cm and its length is twice its breadth. What is its area?

Let the breadth be b, so the length is 2b. Perimeter = 2(l + b) = 2(2b + b) = 6b = 36, so b = 6 cm and l = 12 cm. Area = 12 × 6 = 72 cm².

Two traps sit in this one. Dividing 36 by 2 and calling the result the length ignores that the perimeter counts each side twice. And the answer is an area, so the unit is cm² — an option reading "72 cm" is there for the candidate who stopped one step early.

A room is 5 m long and 4 m wide. What is the cost of carpeting it at ₹120 per square metre?

Area = 5 × 4 = 20 m². Cost = 20 × 120 = ₹2,400. Carpeting is an area question; fencing the same room would be a perimeter one.

A rectangular field 30 m by 20 m has a path 2 m wide running all around it on the outside. Find the area of the path.

Outer rectangle = (30 + 4) × (20 + 4) = 34 × 24 = 816 m². Inner = 600 m². Path = 816 − 600 = 216 m². The width is added twice — once at each end — and forgetting the second is the standard error.

How many cubic centimetres are there in 2 m³?

1 m³ = 1,000,000 cm³, so 2,000,000 cm³. An option of 200,000 is there for anyone who used 100 rather than 100³.

A water tank measures 2 m × 1.5 m × 1 m. How many litres does it hold when full?

Volume = 3 m³, and 1 m³ = 1000 L, so 3,000 litres. Working in centimetres gives 3,000,000 cm³, which is the same answer once divided by 1000 — and an option of 3,000,000 is waiting for whoever forgets to.

The radius of a circle is 14 cm. Its area is — (a) 88 cm² (b) 616 cm² (c) 616 cm (d) 44 cm²

πr² = (22/7) × 14 × 14 = 616 cm². The radius is a multiple of 7 because the examiner chose it so. Option (a) is the circumference, 2πr = 88, and (c) is the right number with the wrong unit.

The sides of a triangle are 13 cm, 14 cm and 15 cm. Its area is —

No height is given, so Heron's formula. s = (13 + 14 + 15) ÷ 2 = 21. Area = √(21 × 8 × 7 × 6) = √7056 = 84 cm².

If every edge of a cube is doubled, its volume becomes — (a) twice (b) four times (c) eight times (d) sixteen times

Volume goes as the cube of the length, so 2³ = 8. The matching question about surface area has the answer four times.

A floor 6 m by 4 m is to be covered with square tiles of side 20 cm. How many tiles are needed?

Floor = 24 m² = 240,000 cm². One tile = 400 cm². 600 tiles. Doing it in mixed units — 24 ÷ 0.2 — produces a confident wrong answer, which is why the unit has to be settled before the division.

The curved surface area of a cylinder of radius 7 cm and height 10 cm is —

2πrh = 2 × (22/7) × 7 × 10 = 440 cm². If the question had asked for the total surface area it would be 2πr(r + h) = 2 × (22/7) × 7 × 17 = 748 cm², and reading which of the two is wanted is half the question.

Mistakes that cost marks here

• Giving an area in cm, or a volume in cm². The commonest avoidable loss in this topic.
• Converting area or volume by 100 instead of 100² and 100³.
• Confusing 1 litre with 1 cm³. One litre is 1000 cm³.
• Adding a path's width once instead of twice when finding the outer rectangle.
• Assuming equal perimeters mean equal areas, or that doubling the sides doubles the area.
• Using the height of a cone where the slant height is wanted, or the reverse.
• Confusing a curved surface area with a total one — the question says which.
• Mixing metres and centimetres in a tiles-and-bricks division instead of converting first.
• Reaching for ½ × base × height when only the three sides are known, where Heron's formula is wanted.
• Using 3.14 where 22/7 was intended, when the radius is a multiple of 7 and the arithmetic was meant to come out whole.

In one line

The formulas are few and fixed; what the paper actually tests is the unit on the answer and the squared and cubed conversions — lengths scale by k, areas by k², volumes by k³, and a litre is a thousand cubic centimetres.

For the classroom

Every formula here was a discovery before it was a formula: perimeter by walking the edge of the classroom, area by covering a desk with square tiles, volume by filling a box with cubes. A child who has filled the box knows why volume is cubed, and will not later write cm² under it.

The squared conversion is worth doing the same way. Draw a square metre on the floor and ask how many hundred-centimetre squares fit along each side; the hundred-by-hundred is visible, and 10,000 stops being a number to remember.

And for the scaling rule, cut a paper square and a second one with sides twice as long, then lay the small one on the big one. Four of them fit. No child who has seen that will ever again say that doubling the side doubles the area.