SPI Mathematics · Mensuration 2D

Rectangles, Squares and Triangles

Perimeter is the distance round a shape and area is the space it covers; for a rectangle they are 2(l + b) and l × b.

Why this matters

Thirteen questions in the bank, two of them from the 2022 paper: ten EASY and three MODERATE. Rounds of a park, carpeting a room, sides in a ratio, the diagonal of a hall and the area of a regular hexagon. Most marks are lost on mixing perimeter with area, or metres with centimetres.

Concept 1 of 3: Perimeter, area and cost of a rectangle

Walk round a field and you cover its perimeter; lay carpet in it and you cover its area. Fencing is paid by the metre of perimeter, carpeting by the square metre of area.

Definition

Perimeter $= 2(l + b)$ ; area $= l \times b$ . For a square of side $a$ : $4a$ and $a^2$ .
Sides in a ratio $m : n$ : write them as $mk$ and $nk$ , then find $k$ from the perimeter or the area.
Cost of carpeting or tiling $=$ area $\times$ rate. Fencing $=$ perimeter $\times$ rate.
At the same rate, cost goes with area. If the length stays fixed, cost goes with the breadth alone.

Rectangle and square

P = 2(l + b), \quad A = lb, \quad P_{\text{square}} = 4a, \quad A_{\text{square}} = a^2

l, blength and breadth
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:

SPI · 2022 · SPI Entrance Examination 2022 · Q19Moderate

Example 1 · Mensuration 2D · Rectangles, Squares and Triangles

The cost of carpeting, 20m long room is Rs40/. Had the breadth been 3m less, the cost would have been Rs25/-. The area of the room is ____sq m.

Perimeter where area is needed

Fencing and walking round use the perimeter; carpeting, tiling and painting use the area. Decide which before you multiply by the rate.

Forgetting to halve the perimeter

A perimeter of 30 cm means

l + b = 15

cm, not 30 cm. Divide by 2 before splitting the sum in a ratio.

Counting rectangles with a fixed perimeter

Infinitely many rectangles have a perimeter of 14 cm. Only if the sides must be whole centimetres is the answer a count:

l + b = 7

gives 1 by 6, 2 by 5 and 3 by 4, so three.

Drill 5 more on perimeter, area and cost of a rectangle

Concept 2 of 3: Diagonals of rectangles and squares

A diagonal cuts a rectangle into two right triangles, with the diagonal as the hypotenuse. Pythagoras connects the two sides and the diagonal.

Definition

Rectangle: $d^2 = l^2 + b^2$ .
Square: $d = a\sqrt{2}$ , so the area is $\frac{d^2}{2}$ .
Put both sides in the same unit before squaring.
Whole-number sets worth knowing: 3, 4, 5; 5, 12, 13; 8, 15, 17; 7, 24, 25 (and their multiples).

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 has length 24 cm and diagonal 25 cm. Find its breadth and its area.

Practice this conceptself-check · 4 quick reps

The same idea in a practice question:

Practice questionEasy

Example 2 · Mensuration 2D · Rectangles, Squares and Triangles

The length and breadth of a rectangle are in the ratio 4 : 3. If the diagonal measures 25 cm then the perimeter of the rectangle is

Mixing metres and centimetres

A hall 0.5 m by 30 cm must become 50 cm by 30 cm before squaring. Options often print the same number in both units, so check the unit too.

Doubling the diagonal doubles the area

It does not. A square's area is

\frac{d^2}{2}

, so twice the diagonal gives four times the area.

Side or perimeter from an area

A square of area 81 square cm has side

\sqrt{81} = 9

cm. Its perimeter, 36 cm, is the usual wrong option.

Drill 3 more on diagonals of rectangles and squares

Concept 3 of 3: Triangles and the regular hexagon

Every triangle is half of a rectangle with the same base and height. A regular hexagon splits from its centre into six equal equilateral triangles.

Definition

Any triangle: area $= \frac{1}{2} \times$ base $\times$ height. In a right triangle the two shorter sides are the base and height.
Equilateral triangle of side $a$ : perimeter $3a$ , area $\frac{\sqrt{3}}{4}a^2$ .
Regular hexagon of side $a$ : six equilateral triangles of side $a$ , so area $= 6 \times \frac{\sqrt{3}}{4}a^2 = \frac{3\sqrt{3}}{2}a^2$ .

Triangle and hexagon

A = \frac{1}{2}bh, \quad A_{\text{equilateral}} = \frac{\sqrt{3}}{4}a^2, \quad A_{\text{hexagon}} = 6 \times \frac{\sqrt{3}}{4}a^2 = \frac{3\sqrt{3}}{2}a^2

Worked example

Find the area of a regular hexagon of side 6 cm.

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

SPI · 2022 · SPI Entrance Examination 2022 · Q69Easy

Example 3 · Mensuration 2D · Rectangles, Squares and Triangles

The area of regular hexagon of side 4m is ____.

Forgetting the half

A triangle's area is half of base times height. Base 10 cm and height 6 cm give 30 square cm, not 60.

Too few triangles in a hexagon

A regular hexagon is six equilateral triangles, so multiply one triangle's area by 6. A hexagon of side 3 cm is

6 \times \frac{9\sqrt{3}}{4} = \frac{27\sqrt{3}}{2}

square cm, not

\frac{9\sqrt{3}}{4}

Drill 2 more on triangles and the regular hexagon

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)

Perimeter, area and cost of a rectangle
Rectangle and square
$P = 2(l + b), \quad A = lb, \quad P_{\text{square}} = 4a, \quad A_{\text{square}} = a^2$
Diagonals of rectangles and squares
Diagonals
$d = \sqrt{l^2 + b^2}, \qquad d_{\text{square}} = a\sqrt{2}, \qquad A_{\text{square}} = \frac{d^2}{2}$
Triangles and the regular hexagon
Triangle and hexagon
$A = \frac{1}{2}bh, \quad A_{\text{equilateral}} = \frac{\sqrt{3}}{4}a^2, \quad A_{\text{hexagon}} = 6 \times \frac{\sqrt{3}}{4}a^2 = \frac{3\sqrt{3}}{2}a^2$

Watch out for (8)

Perimeter where area is needed
→ Perimeter, area and cost of a rectangle
Forgetting to halve the perimeter
→ Perimeter, area and cost of a rectangle
Counting rectangles with a fixed perimeter
→ Perimeter, area and cost of a rectangle
Mixing metres and centimetres
→ Diagonals of rectangles and squares
Doubling the diagonal doubles the area
→ Diagonals of rectangles and squares
Side or perimeter from an area
→ Diagonals of rectangles and squares
Forgetting the half
→ Triangles and the regular hexagon
Too few triangles in a hexagon
→ Triangles and the regular hexagon

Test yourself on a real paper

Sit a past SPI paper, timed and marked the way the exam marks it. You see your score and every answer the moment you finish. Free to start.

Sit an SPI past paper Drill all 13 questions