SSC CGL Maths · Mensuration 2D

Rectangles, Squares and Other Quadrilaterals

A rectangle's area is length times breadth and its perimeter is twice their sum; a square, a parallelogram and a trapezium each have one short rule of their own.

All Mensuration 2D notes

Why this matters

Twenty-five questions in the bank, all but one EASY, seventeen of them from the 2025 shifts. They are quick marks: write the sides as one unknown, use the right rule, and watch the units. Tiles, paths and painting costs are the same rules with one extra step.

Concept 1 of 4: Area and perimeter of a rectangle and a square

Area counts the unit squares inside a shape; perimeter is the distance round its edge. For a rectangle both come straight from its two sides, so most questions are really about finding those two sides from what the stem gives you.

Definition

Rectangle with length $l$ and breadth $b$ : area $lb$ , perimeter $2(l + b)$ .
Square with side $a$ : area $a^2$ , perimeter $4a$ . From the area, the side is $\sqrt{\text{area}}$ .
When the sides are linked (a ratio, or one side '2 m less' than the other), write both sides with one unknown and use the one number the stem gives.
Fast route: half the perimeter is $l + b$ . With a ratio such as $5 : 3$ , split that half in the ratio and the sides fall out with no equation. Safe whenever the sides are given as a ratio and the perimeter is known.
A regular polygon with $n$ equal sides of length $a$ has perimeter $na$ .

Rectangle and square

A = lb \qquad P = 2(l + b) \qquad A = a^2 \qquad P = 4a

l, blength and breadth of the rectangle
aside of the square

Worked example

The length and breadth of a rectangle are in the ratio 5 : 3 and its perimeter is 64 cm. Find its area.

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

SSC CGL · 2025 · Tier I, 26 Sep 2025, Shift 2 · Q68Easy

Example 1 · Mensuration 2D · Rectangles, Squares and Other Quadrilaterals

A rectangle has a length that is four times its width. If the perimeter is 100 cm, what is the area of the rectangle?

Dividing a rectangle's perimeter by 4

Only a square has side

\frac{P}{4}

. For a rectangle,

\frac{P}{2}

is the sum

l + b

, and you still need a second fact to split it.

Multiplying lengths in different units

A length in metres and a breadth in centimetres must be changed to one unit before you multiply.

5\ \text{m} \times 40\ \text{cm}

5 \times 0.4 = 2\ \text{m}^2

, not 200.

Drill 11 more on area and perimeter of a rectangle and a square

Concept 2 of 4: Diagonal of a rectangle and of a square

A diagonal cuts a rectangle into two right triangles, with the length and the breadth as the two shorter sides. So the diagonal is a hypotenuse, and Pythagoras gives it at once.

Definition

Rectangle: diagonal $d = \sqrt{l^2 + b^2}$ .
Square of side $a$ : diagonal $a\sqrt{2}$ .
A square's area from its diagonal: $\frac{d^2}{2}$ . No need to find the side first.
Fast route: know the common right triangles. $3, 4, 5$ ; $5, 12, 13$ ; $8, 15, 17$ ; $7, 24, 25$ and their multiples. A $16 \times 12$ rectangle is the $3, 4, 5$ triangle scaled by 4, so its diagonal is 20. Safe only when both sides are the same multiple of a known triple; otherwise square and add.

Diagonals

d = \sqrt{l^2 + b^2} \qquad d_{\text{square}} = a\sqrt{2} \qquad A_{\text{square}} = \frac{d^2}{2}

Worked example

A rectangle is 16 cm long and 12 cm wide. Find the length of its diagonal.

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

SSC CGL · 2025 · Tier I, 23 Sep 2025, Shift 2 · Q60Easy

Example 2 · Mensuration 2D · Rectangles, Squares and Other Quadrilaterals

A square field has a diagonal of

\sqrt{98}

m. What is its area?

Squaring the diagonal for a square's area

The square of the diagonal is twice the area. A square with diagonal 8 m has area

\frac{64}{2} = 32\ \text{m}^2

, not 64.

Adding the sides to get the diagonal

The diagonal of a rectangle is

\sqrt{l^2 + b^2}

, always shorter than

l + b

. A 6 by 8 rectangle has diagonal 10, not 14.

Drill 3 more on diagonal of a rectangle and of a square

Concept 3 of 4: Tiles, paths, cut-outs and the cost of covering

These are word problems wrapped round one area. Find the area that is actually covered (a wall minus its window, a lawn plus its path), then divide by one tile's area or multiply by the rate.

Definition

Tiles needed $=$ area to cover $\div$ area of one tile. When each side divides exactly by the tile's side, count along each side and multiply; that is also the check.
Extra tiles for breakage: multiply by $1 + \frac{p}{100}$ and round up, since part of a tile cannot be bought.
Path of width $w$ outside an $l \times b$ rectangle: the outer rectangle is $(l + 2w) \times (b + 2w)$ ; the path is outer minus inner.
Path inside the edge: the inner rectangle is $(l - 2w) \times (b - 2w)$ .
Cut-out: subtract the removed area.
Cost: painting, paving or tiling is charged per square metre (area $\times$ rate); fencing is charged per metre (perimeter $\times$ rate).
Fast route for an outside path: its area is $2w(l + b) + 4w^2$ , with no need to work out the outer rectangle. Safe only for a path of the same width on every side.

Tiles and paths

N = \frac{\text{area to cover}}{\text{area of one tile}} \qquad A_{\text{path}} = (l + 2w)(b + 2w) - lb = 2w(l + b) + 4w^2

wwidth of the path
Nnumber of tiles

Worked example

A rectangular lawn 20 m by 15 m has a path 2 m wide all round it, outside the lawn. Find the area of the path and the cost of paving it at ₹50 per m².

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

SSC CGL · 2025 · Tier I, 24 Sep 2025, Shift 2 · Q58Easy

Example 3 · Mensuration 2D · Rectangles, Squares and Other Quadrilaterals

A floor is 12 m by 9 m. If tiles of 1.5 m

\times

1.5 m are used, how many tiles are needed?

Adding the path width only once

A path on every side adds its width at both ends, so each side of the outer rectangle grows by

2w

, not

w

Rounding the tile count down

If the working gives 89.6 tiles, buy 90. A count of tiles, boxes or trips is always rounded up to cover the whole job.

Charging fencing by area

Fencing runs along the boundary, so its cost is perimeter times the rate per metre. Painting, paving and tiling cover the surface, so they use the area.

Drill 4 more on tiles, paths, cut-outs and the cost of covering

Concept 4 of 4: Area of a parallelogram, a trapezium and a rhombus

Slide the slanted end of a parallelogram across and it becomes a rectangle with the same base and height. A trapezium is a rectangle whose width is the average of its two parallel sides.

Definition

Parallelogram: area $=$ base $\times$ height, where the height is the perpendicular distance between the base and the opposite side, not the slanting side.
Trapezium with parallel sides $a$ and $b$ a distance $h$ apart: area $= \frac{1}{2}(a + b)h$ .
Rhombus with diagonals $d_1$ and $d_2$ : area $= \frac{1}{2}d_1 d_2$ .
When the parallel sides are given as a ratio, write them as $3x$ and $5x$ and solve one equation.
A plot made of several pieces: find each area and add.

Parallelogram, trapezium and rhombus

A = bh \qquad A = \frac{1}{2}(a + b)h \qquad A = \frac{1}{2}d_1 d_2

a, bthe two parallel sides of a trapezium
hperpendicular height
$d_1, d_2$ diagonals of a rhombus

Worked example

A trapezium has parallel sides of 14 cm and 22 cm, and the distance between them is 9 cm. Find its area.

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

SSC CGL · 2025 · Tier I, 18 Sep 2025, Shift 2 · Q69Easy

Example 4 · Mensuration 2D · Rectangles, Squares and Other Quadrilaterals

In a trapezoid, the two parallel sides are in the ratio 3:5. If the height of the trapezoid is 10 cm and its total area is

240\ \text{cm}^2

, determine the lengths of the parallel sides.

Using the slanting side as the height

A parallelogram's area uses the perpendicular height. A slanting side of 10 cm with a perpendicular height of 8 cm on a base of 12 cm gives

96\ \text{cm}^2

, not 120.

Forgetting the half in a trapezium

A trapezium's area is the AVERAGE of the parallel sides times the height. Adding the sides and multiplying by the height doubles the answer.

Drill 3 more on area of a parallelogram, a trapezium and a rhombus

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 (4)

Area and perimeter of a rectangle and a square
Rectangle and square
$A = lb \qquad P = 2(l + b) \qquad A = a^2 \qquad P = 4a$
Diagonal of a rectangle and of a square
Diagonals
$d = \sqrt{l^2 + b^2} \qquad d_{\text{square}} = a\sqrt{2} \qquad A_{\text{square}} = \frac{d^2}{2}$
Tiles, paths, cut-outs and the cost of covering
Tiles and paths
$N = \frac{\text{area to cover}}{\text{area of one tile}} \qquad A_{\text{path}} = (l + 2w)(b + 2w) - lb = 2w(l + b) + 4w^2$
Area of a parallelogram, a trapezium and a rhombus
Parallelogram, trapezium and rhombus
$A = bh \qquad A = \frac{1}{2}(a + b)h \qquad A = \frac{1}{2}d_1 d_2$

Watch out for (9)

Dividing a rectangle's perimeter by 4
→ Area and perimeter of a rectangle and a square
Multiplying lengths in different units
→ Area and perimeter of a rectangle and a square
Squaring the diagonal for a square's area
→ Diagonal of a rectangle and of a square
Adding the sides to get the diagonal
→ Diagonal of a rectangle and of a square
Adding the path width only once
→ Tiles, paths, cut-outs and the cost of covering
Rounding the tile count down
→ Tiles, paths, cut-outs and the cost of covering
Charging fencing by area
→ Tiles, paths, cut-outs and the cost of covering
Using the slanting side as the height
→ Area of a parallelogram, a trapezium and a rhombus
Forgetting the half in a trapezium
→ Area of a parallelogram, a trapezium and a rhombus

Test yourself on Mensuration 2D

20 past SSC CGL questions from this chapter, timed at 12 minutes and marked the way the exam marks it. You see your score and every answer the moment you finish. Free to start.

Take the Mensuration 2D chapter test Drill all 25 questions