AFCAT Numerical Ability · Area and Perimeter

Rectangles and Squares: Area, Perimeter, Diagonal and Cost

Area is length × breadth, perimeter is twice their sum and the diagonal comes from Pythagoras; most questions give two facts and ask you to find the sides first.

All Area and Perimeter notes

Why this matters

Every other figure in the chapter is measured against these formulas, so the page is mostly EASY. Marks go on two slips: mixing up the perimeter with the area, and dividing the perimeter in a ratio before halving it. The cost, tile and path questions are the same formulas with one extra step.

Concept 1 of 3: Area, perimeter and diagonal of a rectangle and a square

Area counts the unit squares that cover a shape; perimeter is the distance round its edge. A rectangle

l

b

holds

l \times b

unit squares and has an edge of

l + b + l + b

. Its diagonal is the hypotenuse of a right triangle with legs

l

and

b

, so it comes from Pythagoras, which the Triangles page takes up properly.

Definition

Rectangle: area $lb$ , perimeter $2(l + b)$ , diagonal $\sqrt{l^2 + b^2}$ .
Square of side $a$ : area $a^2$ , perimeter $4a$ , diagonal $a\sqrt 2$ .
Square from its diagonal $d$ : area $\dfrac{d^2}{2}$ , because the side is $\dfrac{d}{\sqrt 2}$ .
Area is in square units (cm², m²); perimeter and diagonal are lengths (cm, m).

Rectangle and square

A = lb,\quad P = 2(l + b),\quad d = \sqrt{l^2 + b^2};\qquad A_{\text{square}} = a^2 = \dfrac{d^2}{2}

Worked example

A rectangle is 24 m long and 7 m wide. Find its area, perimeter and diagonal.

Practice this conceptself-check · 4 quick reps

The same idea in a practice question:

Practice questionEasy

Example 1 · Area and Perimeter · Rectangles and Squares

The diagonal of a square field measures 50 m. The area of square field is

A square from its diagonal

A square with a diagonal of 6 cm has an area of

\tfrac{36}{2} = 18

cm², not 36 cm². The diagonal is longer than the side by a factor of

\sqrt 2

Perimeter is a length

A 9 cm by 4 cm rectangle has area 36 cm² and perimeter 26 cm. Check the unit an option carries: cm² means an area was asked.

Drill 6 more on area, perimeter and diagonal of a rectangle and a square

Concept 2 of 3: Finding the sides from two facts

A rectangle has two unknowns, its length and its breadth, so it needs two facts. The perimeter gives their sum; a ratio, a difference, the area or the diagonal gives the second equation.

Definition

Half the perimeter is $l + b$ . Halve first, then split it in the ratio or use the difference.
Sum and diagonal: $(l + b)^2 - (l^2 + b^2) = 2lb$ , so the area comes out without finding $l$ and $b$ .
Two rectangles of equal area: $l_1 b_1 = l_2 b_2$ .
A shape fact alone ('the length is twice the breadth') fixes no size. Statement questions turn on this: one fact about shape and one about size are needed together.
With $a = lb$ and $p = 2(l + b)$ , any claimed identity can be tested by substituting both.

Two facts, two sides

l + b = \dfrac{P}{2},\qquad 2lb = (l + b)^2 - d^2

Worked example

A rectangle has a perimeter of 56 cm and its length and breadth are in the ratio

4 : 3

. Find its area and its diagonal.

Practice this conceptself-check · 4 quick reps

The same idea in a practice question:

Practice questionEasy

Example 2 · Area and Perimeter · Rectangles and Squares

The perimeter of a rectangular field is 240 m and the ratio between the length and breadth is 5 : 3. The area of the field is

Halve the perimeter before splitting it

With a perimeter of 56 and a ratio of

4 : 3

, split 28, not 56. Splitting the full perimeter doubles both sides and makes the area four times too big.

Shape is not size

'The length is three times the breadth' is true of a 3 by 1 rectangle and a 30 by 10 one. A statement like that needs a second fact about size before any area follows.

Drill 4 more on finding the sides from two facts

Concept 3 of 3: Fencing, flooring, tiles and paths

A cost question asks which measure is being paid for. Fencing and borders go round the edge, so they use the perimeter; carpeting, paving and painting cover the surface, so they use the area. A path is the difference between two rectangles.

Definition

Cost = perimeter × rate per metre (fencing), or area × rate per m² (flooring).
Tiles = area of the floor ÷ area of one tile, both in the same unit.
Fewest square slabs that fit exactly: the slab side is the HCF of the length and the breadth.
Path of width $w$ outside an $l$ by $b$ plot: $(l + 2w)(b + 2w) - lb$ . Inside: $lb - (l - 2w)(b - 2w)$ .
Two crossing paths through the middle: $w(l + b) - w^2$ , the square where they cross counted once.
When the path's width is the unknown, the area left gives a quadratic. Keep only the root that fits inside the plot.

Path outside a rectangle

\text{path} = (l + 2w)(b + 2w) - lb = 2w(l + b) + 4w^2

Worked example

A lawn 30 m by 18 m has a path 3 m wide all round it, outside. Find the area of the path, and the cost of paving it at ₹15 per m².

Practice this conceptself-check · 4 quick reps

The same idea in a practice question:

Practice questionModerate

Example 3 · Area and Perimeter · Rectangles and Squares

Within a rectangular garden 10 m wide and 20 m long, we wish to pave a walk around the borders of uniform width so as to leave an area of

96 \text{ m}^2

for flowers. The width of the walk is

A path adds its width on both sides

A path 3 m wide round a 30 m lawn makes the outer length 36 m, not 33 m. The width is added at each end.

Same units before dividing

A 6 m by 4 m floor is 24 m², and a 40 cm tile is 0.16 m². Dividing 24 by 1,600 (the tile in cm²) gives nonsense.

Throw out the root that cannot fit

A quadratic for a path's width has two roots. A width that is more than half the plot's breadth would leave nothing inside, so it is rejected.

Drill 3 more on fencing, flooring, tiles and paths

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (3)

Area, perimeter and diagonal of a rectangle and a square
Rectangle and square
$A = lb,\quad P = 2(l + b),\quad d = \sqrt{l^2 + b^2};\qquad A_{\text{square}} = a^2 = \dfrac{d^2}{2}$
Finding the sides from two facts
Two facts, two sides
$l + b = \dfrac{P}{2},\qquad 2lb = (l + b)^2 - d^2$
Fencing, flooring, tiles and paths
Path outside a rectangle
$\text{path} = (l + 2w)(b + 2w) - lb = 2w(l + b) + 4w^2$

Watch out for (7)

A square from its diagonal
→ Area, perimeter and diagonal of a rectangle and a square
Perimeter is a length
→ Area, perimeter and diagonal of a rectangle and a square
Halve the perimeter before splitting it
→ Finding the sides from two facts
Shape is not size
→ Finding the sides from two facts
A path adds its width on both sides
→ Fencing, flooring, tiles and paths
Same units before dividing
→ Fencing, flooring, tiles and paths
Throw out the root that cannot fit
→ Fencing, flooring, tiles and paths

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