Area and Perimeter is a high-yield AFCAT topic because the formulas are fixed and the questions are direct — once you know the shape, you know the answer. This Cavalier guide collects every plane-figure formula you need, then layers on the scaling rule, the diagonal trick and the path-around-a-field method that examiners reuse year after year. Get these down and a whole cluster of marks becomes nearly automatic.
Area versus perimeter: the core distinction
Two ideas run through this entire topic. Perimeter is the distance once around the boundary of a flat figure — a length, measured in metres or centimetres. Area is the amount of surface the figure covers — a square measure, in square metres or square centimetres.
Almost every AFCAT mistake here comes from confusing the two: fencing a field uses perimeter, but turfing or tiling it uses area; the cost of a wire around a plot is perimeter, the cost of levelling it is area. Read the question for the operation — ‘around’, ‘border’ and ‘fence’ signal perimeter, while ‘cover’, ‘tile’, ‘paint’ and ‘turf’ signal area.
At The Cavalier we tell defence aspirants to treat Area and Perimeter as guaranteed marks. There is no clever theory to derive; success is purely about clean formula recall and careful unit handling. The candidates who lose marks here are the ones who rush the units or pick the wrong measure — not the ones who forgot a formula.
Perimeter is a single power of length (a line). Area is a squared power of length (a surface). Whenever you double-check an answer, confirm the units match the quantity asked for.
Rectangle and square
These two shapes account for the largest share of AFCAT questions, so their formulas must be instant.
- Rectangle: Area = length × breadth; Perimeter = 2 × (length + breadth)
- Rectangle diagonal = √(length2 + breadth2)
- Square: Area = side2; Perimeter = 4 × side
- Square in terms of diagonal d: Area = d2 ÷ 2; diagonal = side × √2
A rectangular plot is 40 m long and 30 m wide. Find its area, perimeter and diagonal.
If a question gives only the diagonal of a square, use Area = d2 ÷ 2 directly — there is no need to find the side first.
Triangle: every formula you need
Triangles appear in several guises, so keep all the area routes ready and pick the one that matches the given data.
- General: Area = ½ × base × height
- Equilateral (side a): Area = (√3 ÷ 4) × a2; height = (√3 ÷ 2) × a
- Right triangle: Area = ½ × (the two legs)
- Three sides given (Heron's): s = (a + b + c) ÷ 2, Area = √[s(s − a)(s − b)(s − c)]
Find the area of an equilateral triangle of side 8 cm.
Find the area of a triangle with sides 13, 14 and 15 cm using Heron's formula.
The perimeter of any triangle is just the sum of its three sides. For an equilateral triangle that is 3 × side.
Parallelogram, rhombus and trapezium
These quadrilaterals share family resemblances but each has its own area route. Keep the diagonal-based rhombus formula handy — it is a frequent AFCAT favourite.
- Parallelogram: Area = base × height; Perimeter = 2 × (sum of adjacent sides)
- Rhombus: Area = ½ × d1 × d2 (product of diagonals); Perimeter = 4 × side
- Rhombus side from diagonals = √[(d1/2)2 + (d2/2)2]
- Trapezium: Area = ½ × (sum of the two parallel sides) × height
A rhombus has diagonals 24 cm and 10 cm. Find its area and perimeter.
The diagonals of a rhombus bisect each other at right angles, so halve each diagonal before using Pythagoras to find the side. Forgetting to halve is the usual slip.
Circle: area, circumference and sectors
The circle brings in π (taken as 22/7 or 3.14 on AFCAT). Keep both the area and circumference formulas instant, and know the semicircle and sector versions too.
- Area = π × r2; Circumference = 2 × π × r = π × d
- Semicircle: Area = ½ π r2; Perimeter = π r + 2r
- Sector of angle θ: Area = (θ ÷ 360) × π r2; Arc length = (θ ÷ 360) × 2π r
- Ring (annulus) between radii R and r: Area = π(R2 − r2)
A circular park has radius 14 m. Find its area and circumference (take π = 22/7).
Choose π = 22/7 whenever the radius is a multiple of 7, and π = 3.14 otherwise. Matching π to the radius keeps the arithmetic clean.
The scaling rule: how area responds to a change in size
This is the single most tested concept in the topic. When every linear dimension of a figure is multiplied by a factor k, the perimeter scales by k but the area scales by k2. This is because area is a squared measure.
If each side (or the radius) changes by a factor k, then Perimeter → k × old, but Area → k2 × old.
The side of a square is increased by 20%. By what per cent does its area increase?
So a 20% rise in side gives a 44% rise in area, not 40%. The same logic works in reverse: if a circle's area is to be quadrupled, the radius must only double, since 22 = 4. Recognising the squared relationship lets you skip the full calculation and answer percentage-change questions almost instantly.
Never apply the linear percentage straight to area. A 10% increase in radius raises area by 21% (1.12 = 1.21), not 10%.
Paths, borders and roads around or through a field
A recurring AFCAT setup places a uniform path around a garden, or two roads crossing a rectangular field. The method is always to find the difference of two areas, or to add the two roads and remove the double-counted overlap.
Path around the outside: outer dimensions = field + 2w on each side, where w is the path width. Path area = outer area − field area. Path inside the field subtracts 2w instead.
A rectangular lawn 50 m by 30 m has a 2 m wide path running all around it on the outside. Find the area of the path.
For two roads crossing the middle of a field, Road area = (length road + breadth road) − the square overlap where they cross, otherwise that square is counted twice.
Cost of fencing, turfing and tiling
Real-world cost questions simply attach a rate to either the perimeter or the area. The skill is choosing the right measure first, then multiplying by the rate.
Cost of fencing or wiring = Perimeter × rate per metre. Cost of turfing, tiling, paving or levelling = Area × rate per square metre. Identify ‘around’ versus ‘cover’ before you calculate.
A square field of side 25 m is to be fenced at 12 per metre and turfed at 8 per square metre. Find each cost.
For tiling, divide the floor area by the tile area to get the number of tiles — do not divide perimeters or mix square and linear units.
Previous-year style question
Q. The length of a rectangle is increased by 25% while its breadth is decreased by 20%. What is the percentage change in its area?
Answer: New area = old × (1.25) × (0.80) = old × 1.00. The two changes cancel exactly, so the area stays the same — a 0% change. (Using the successive-change formula: 25 + (−20) + (25 × −20 ÷ 100) = 5 − 5 = 0%.)
Traps and time-savers to keep in mind
- Applying a linear percentage change directly to area instead of squaring the factor.
- Mixing units — convert everything to the same unit before calculating.
- Forgetting to halve the rhombus diagonals before using Pythagoras.
- Confusing fencing (perimeter) with turfing or tiling (area).
When a square and a circle, or two shapes, have the same perimeter, the one closer to a circle always encloses the larger area. The circle is the most area-efficient shape for a given boundary — useful for quick comparison questions.
Unit conversion saves marks
Land questions often mix metres with hectares or ares. Keep the conversions ready: 1 hectare = 10,000 square metres and 1 are = 100 square metres. A great many ‘easy’ field questions are missed only because the candidate forgot to convert the final area into the unit the option list uses. Convert first, calculate second, and check the unit of the answer against the options before you commit.
Quick revision
- Perimeter is a length; area is a squared measure — never mix them up.
- Square: Area = side2 = d2/2; Rectangle diagonal = √(l2 + b2).
- Equilateral triangle Area = (√3/4)a2; Heron's for three sides; rhombus Area = ½ d1d2.
- Circle: Area = πr2, Circumference = 2πr; take π = 22/7 when r is a multiple of 7.
- Scaling rule: side ×k gives perimeter ×k but area ×k2.
- Path area = outer area − inner area; fencing uses perimeter, turfing uses area.
Frequently asked questions
How many Area and Perimeter questions come in AFCAT?
Usually one or two direct questions per paper, and the formulas also support Mensuration and Data Interpretation items. Because the questions are formula-driven, it is one of the most reliable scoring areas.
Why does a 20% increase in side raise the area by 44%, not 40%?
Area depends on the square of the side. Multiplying the side by 1.20 multiplies the area by 1.20 squared, which is 1.44 — a 44% increase. Always square the scaling factor for area.
When should I use 22/7 and when 3.14 for pi?
Use 22/7 when the radius or diameter is a multiple of 7, because the sevens cancel cleanly. Use 3.14 otherwise. Matching pi to the number keeps the arithmetic simple and fast.
How do I find the area of a rhombus when only the diagonals are given?
Use Area = half the product of the diagonals, that is half of d1 times d2. The diagonals of a rhombus cross at right angles, which is exactly why this neat formula works.
What is the quickest way to handle a path around a field?
Find the outer area by adding twice the path width to each dimension, then subtract the original field area. The difference is the path area — no need to split it into strips.
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