AFCAT Numerical Ability · Area and Perimeter

Change in Dimensions: Scaling and Percentage Changes

When every length is multiplied by k, perimeters are multiplied by k and areas by k²; percentage changes in two sides multiply, they do not add.

All Area and Perimeter notes

Why this matters

No question here is HARD, yet the page loses marks steadily, because adding two percentages or forgetting to square a ratio gives an answer that is always printed among the options. Each question is one multiplication of factors once the change is written as a factor.

Concept 1 of 2: Scaling a figure and comparing areas

Make every length of a figure k times as long and each unit square inside it becomes a square k times as wide and k times as tall. So lengths, perimeters and radii go up k times, but areas go up k² times. The same holds for any two figures of the same shape.

Definition

Same shape (two squares, two circles, two similar triangles): ratio of perimeters = ratio of sides; ratio of areas = (ratio of sides)².
From areas back to sides: take square roots. Areas $36 : 49$ mean sides $6 : 7$ .
'The area becomes 25 times' is a rise of 24 times the old area, or 2400%. Read whether the question asks for the new area or the rise.
Fixed perimeter: of all rectangles with the same perimeter, the square has the largest area.
Same area, different shape: set the two area formulas equal and solve.

Scale factor k

\dfrac{P_2}{P_1} = k,\qquad \dfrac{A_2}{A_1} = k^2

Worked example

Two circles have radii in the ratio

5 : 7

. Find the ratio of their circumferences and of their areas.

Practice this conceptself-check · 3 quick reps

The same idea in a practice question:

Practice questionModerate

Example 1 · Area and Perimeter · Change in Dimensions

The ratio of areas of two similar triangles is

4:9

. What is the ratio of their corresponding sides?

Square the ratio for area

Radii

5 : 7

give areas

25 : 49

. Keeping

5 : 7

for the areas is the option most often wrongly picked.

Square roots on the way back

Areas

36 : 49

mean sides

6 : 7

. Giving

36 : 49

for the sides, or squaring again, are the usual wrong options.

Drill 6 more on scaling a figure and comparing areas

Concept 2 of 2: Percentage changes in the sides

An area is a product, so changes in its two lengths multiply. A rise of 30% in the length is a factor of 1.3 and a rise of 10% in the breadth a factor of 1.1; the area is multiplied by

1.3 \times 1.1

, and the extra 3% comes from the corner where both strips meet.

Definition

Write each change as a factor: +30% is 1.3, −20% is 0.8. The area's factor is their product.
Changes of $a\%$ and $b\%$ in two sides change the area by $a + b + \dfrac{ab}{100}$ percent (a fall counts as negative).
A square's side or a circle's radius up $x\%$ : both lengths change together, so the area changes by $2x + \dfrac{x^2}{100}$ percent.
To keep the area unchanged after one side rises $x\%$ , the other must fall $\dfrac{x}{100 + x} \times 100$ percent.
Land is often given in hectares: 1 hectare is 10,000 m², a square of side 100 m. Turn it into a side before applying the change.

Two percentage changes

\text{area change} = a + b + \dfrac{ab}{100}\ \%

Worked example

A rectangle's length rises by 30% and its breadth by 10%. By what percentage does its area rise?

Practice this conceptself-check · 4 quick reps

The same idea in a practice question:

Practice questionEasy

Example 2 · Area and Perimeter · Change in Dimensions

The length of a rectangle is increased by 60%. By what per cent would the width have to be decreased to maintain the same area?

Percentages multiply, they do not add

Both sides of a rectangle up 20% raise the area by 44%, not 40%:

1.2 \times 1.2 = 1.44

Undoing a rise is a smaller fall

After a 25% rise in the length, the breadth must fall 20%, not 25%:

1.25 \times 0.8 = 1

Drill 4 more on percentage changes in the sides

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 (2)

Scaling a figure and comparing areas
Scale factor k
$\dfrac{P_2}{P_1} = k,\qquad \dfrac{A_2}{A_1} = k^2$
Percentage changes in the sides
Two percentage changes
$\text{area change} = a + b + \dfrac{ab}{100}\ \%$

Watch out for (4)

Square the ratio for area
→ Scaling a figure and comparing areas
Square roots on the way back
→ Scaling a figure and comparing areas
Percentages multiply, they do not add
→ Percentage changes in the sides
Undoing a rise is a smaller fall
→ Percentage changes in the sides

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