Percentage Change in Area and Volume

📑 On this page

⚡ The short answer

Multiply every length by k and the perimeter becomes × k, the area × k² and the volume × k³. So sides +10% means area +21% and volume +33.1%. Unequal changes? Multiply the factors, or use a + b + ab/100. The traps: answering k instead of k², and treating perimeter like area.

🎬 The 30-second story

The poster that cost four times as much

“Fest poster is 2 ft × 3 ft, ₹240 to print. I'm doubling both sides. Budget: ₹480.”

🖨️

Aarav

Printing is ₹40 per square foot.

“4 ft × 6 ft is 24 sq ft. That's ₹960.”

📐

Zoya

“How? I only doubled it!”

💸

Aarav

The border ribbon did double: 10 ft to 20 ft.

“You doubled it twice. Once across, once down. Four old posters fit inside the new one.”

🧩

Meher

Tara's maths: Lengths (the ribbon) scale by k = 2. Area (the paper, the ink, the bill) scales by k² = 4: from 6 to 24 sq ft, ₹240 to ₹960. Aarav's budget was half what he needed.

Why does doubling a side make the area four times bigger?

Area is a length times a length. Stretch the shape and both lengths stretch, so the factor shows up twice. Volume multiplies three lengths, so it shows up three times. That is the whole topic.

🥧 The split

The doubled poster, split up

The original poster6 sq ft
Extra paper from doubling18 sq ft

The old 2 × 3 poster fills only a quarter of the new 4 × 6 one.

Shortcut: look at the unit. Perimeter is in m (k¹), area in m² (k²), volume in m³ (k³). The power on the unit is the power on the factor.

📐 The figure

Same shape, every length doubled

Small tile

Perimeter 12 · Area 9

the starting square

Big tile

Perimeter 24 · Area 36

perimeter × 2, area × 4

Cube, edge doubled

Volume 8 → 64

volume × 8, surface × 4

Figures are not to scale.

Every length	Perimeter	Area	Volume
× 2	× 2	× 4	× 8
× 3	× 3	× 9	× 27
× ½	× ½	× ¼	× ⅛
+10%	+10%	+21%	+33.1%
−10%	−10%	−19%	−27.1%
+20%	+20%	+44%	+72.8%

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📌 A length is any length

Radius, diameter, diagonal, slant height, a phone's screen size: all lengths, all × k. Surface area is an area, so × k², even on a solid. Only volume, and what follows it (the weight of a solid laddoo), takes × k³.

How do you find the percentage change in area and volume?

Turn each percent into a multiplier and multiply. A 10% rise is × 1.1, so the area goes × 1.1 × 1.1 = 1.21: up 21%. If you prefer to stay in percent, it is the successive-change rule: a + b + ab/100. For the same change x on both sides, that is 2x + x²/100. A 10% cut gives −20 + 1 = −19%.

📊 The data

Every side up by the same percent: how far area and volume run ahead

LengthAreaVolume

Sides +10%

10%

21%

33.1%

Sides +20% · 1.2² = 1.44, 1.2³ = 1.728

20%

44%

72.8%

Sides +50%

50%

125%

237.5%

The bigger the stretch, the wider the gap. Volume runs the rule a third time: 21 + 10 + 2.1 = 33.1.

When only one dimension changes

Now the shape is not a scaled copy, so ask: how does the formula use that length? A rectangle uses l and b once each, so length +20% and breadth −20% gives 20 − 20 − 4 = −4%, not zero. A cylinder's volume πr²h uses r twice and h once.

⚔️ Side by side

Cylinder tank: which change matters more?

↔️Double the radius

r appears squared

•Volume × 2² = × 4
•Curved surface 2πrh: × 2

↕️Double the height

h appears once

•Volume × 2
•Curved surface 2πrh: × 2

Test	Double the radius	Double the height
Same idea in	cones (r²), spheres (r³)	cuboids, prisms

Count how many times the changed length appears in the formula; raise its factor to that power.

Scaling questions solved: warm-up to tricky

✍️ Worked example Warm-up ⏱ 30 sec

The bigger phone screen

Meher upgrades from a 6-inch phone to a 7.5-inch phone with the same screen shape. Screen sizes are diagonals. How much more screen area does she get?

1
A diagonal is a length, so k = 7.5 ÷ 6 = 1.25.
2
Area × k² = 1.25² = 1.5625.

Answer: 56.25% more screen, not 25%

⚡ The exam-hall shortcut

25 + 25 + 25×25/100 = 25 + 25 + 6.25 = 56.25%.

✍️ Worked example CLAT level ⏱ 45 sec

The rooftop water tank

Kabir's cylindrical tank holds 2,000 litres. The new model has a radius 10% larger and a height 10% smaller. What does it hold?

1
V = πr²h. The radius is squared: × 1.1² = × 1.21.
2
The height enters once: × 0.9.
3
Total: 1.21 × 0.9 = 1.089, so 2,000 × 1.089 = 2,178 litres.

Answer: 2,178 litres (+8.9%)

⚡ The exam-hall shortcut

In percent: 21 − 10 − 2.1 = +8.9%. The two 10%s don't cancel, because r counts twice.

✍️ Worked example CLAT level ⏱ 60 sec

Re-flooring the fest hall

Flooring a rectangular hall cost ₹48,000. The hall is extended 20% in length but loses 10% of its breadth, and the rate per m² goes up 5%. What is the new cost?

1
Area factor: 1.2 × 0.9 = 1.08 (area +8%).
2
Cost = area × rate, so multiply by 1.05: 1.08 × 1.05 = 1.134.
3
₹48,000 × 1.134 = ₹54,432.

Answer: ₹54,432 (+13.4%)

⚡ The exam-hall shortcut

Chain the rule: 20 − 10 − 2 = 8; then 8 + 5 + 0.4 = 13.4%. 13.4% of 48,000 is 6,432.

✍️ Worked example Tricky ⏱ 60 sec

The gift box, worked backwards

Rohan's cube-shaped gift box is replaced by a bigger cube that needs 44% more wrapping paper. By what percent has its volume grown?

1
Wrapping paper is surface area, an area: k² = 1.44.
2
So the edge factor is k = √1.44 = 1.2 (edge +20%).
3
Volume × k³ = 1.2³ = 1.728.

Answer: 72.8% more volume

⚡ The exam-hall shortcut

Area 1.44 → length 1.2 → volume 1.728. Step back to the length, then forward.

What is the fastest way to crack a scaling question?

1
Find the length factor
+20% → 1.2, −10% → 0.9, doubled → 2.
2
Count the powers
Perimeter 1, area or surface 2, volume 3. A single changed length takes its power in the formula (r in πr²h → 2).
3
Multiply, then convert
Multiply the factors, subtract 1, × 100: 1.089 → +8.9%.
4
Check the options
An option equal to the length change is usually the trap.

🎛️ Try it yourself — change any number

Stretch every side of the box

Change in every side of the box (%)

Perimeter changes by

+10%

Area changes by

+21%

Volume changes by

+33.1%

Area × (1.1)² ; Volume × (1.1)³

At +10%: area +21%, volume +33.1%. Type 100 (doubling): +300% and +700%. Type −50: −75% and −87.5%. Type 41.42: the area just about doubles, so 'area doubled' means sides × √2.

How does scaling hide inside a CLAT caselet?

CLAT will not say 'scaling'. It will describe an old plan and a revised plan for a park, a hall or a tank, and ask about cost, capacity or fencing. Build a before-and-after table first. Here: a park planned at 120 m × 80 m is revised to length +25% and breadth −10%.

Measure	Old plan	Revised plan	Change
Length	120 m	150 m	+25%
Breadth	80 m	72 m	−10%
Area (turf)	9,600 m²	10,800 m²	+12.5%
Perimeter (fence)	400 m	444 m	+11%

→ scroll to see more

📊 The data

The same two changes, two different answers

Area change · 1.25 × 0.9 = 1.125

12.5%

Perimeter change (120 × 80 park) · 444 ÷ 400

11%

Perimeter change (100 × 100 park) · 430 ÷ 400

7.5%

Area depends only on the percentages. Perimeter depends on the actual sides: same +25% and −10%, different starting shape, different answer.

Which scaling traps cost the most marks?

🧭 Decision flow

Which rule do I use?

❓ Did every length change by the same factor k?

Yes →

❓ What is asked?

Perimeter or any length →

Multiply by k

Area or surface area →

Multiply by k²

Volume or capacity →

Multiply by k³

No →

❓ Is the quantity a product, like area or volume?

Yes →

Multiply each factor, raised to its power in the formula

r in πr²h counts twice

No ↓

❓ Are the actual sides given?

Yes →

Work out old and new perimeter directly

No →

Cannot be determined

perimeter needs real lengths

⚠️ Area doubled does not mean side doubled

Going backwards, take the square root. Area × 2 means each side × √2 ≈ 1.414: up about 41.4%, not 100% and not 50%. Area up 50% means sides up √1.5 − 1 ≈ 22.5%. Volume × 8 means edge × 2 (cube root).

How would CLAT ask it? Three CLAT-style caselets

🧪 CLAT-style practice (not a past question) Area change × rate

A city council first planned a rectangular park 120 m long and 80 m wide, with a cylindrical underground water tank of radius 5 m and depth 4 m. In the revised plan, the park's length is increased by 25% and its breadth reduced by 10%. The tank's radius is increased by 20% and its depth reduced by 20%. Turf costs ₹150 per m² under both plans.

How much more will turfing the revised park cost than the original?

A₹1,44,000

B₹1,80,000

C₹2,16,000

D₹3,60,000

Reveal answer + reasoning

Answer: B

Area factor = 1.25 × 0.9 = 1.125, so the area rises 12.5%. Old area = 120 × 80 = 9,600 m²; extra = 12.5% of 9,600 = 1,200 m² (check: 150 × 72 = 10,800). Extra cost = 1,200 × 150 = ₹1,80,000. About 50 seconds.

Why the others fail

A — Uses only the 10% breadth change, 960 m².
C — Adds 25 − 10 = 15% and drops the cross term (−2.5%).
D — Uses only the 25% length increase and ignores the breadth cut.

🧪 CLAT-style practice (not a past question) One length squared

Same passage as above: the tank's radius is increased by 20% and its depth reduced by 20%.

By what percent does the tank's capacity change?

ANo change

B4% decrease

C15.2% increase

D44% increase

Reveal answer + reasoning

Answer: C

Volume = πr²h. The radius is squared: 1.2² = 1.44. Depth: × 0.8. Total 1.44 × 0.8 = 1.152, so +15.2% (about 314 m³ to 362 m³). In percent: 44 − 20 − 44×20/100 = 15.2. About 40 seconds.

Why the others fail

A — Assumes +20% and −20% cancel.
B — Treats it like a rectangle (r used once): 20 − 20 − 4.
D — Squares the radius but forgets the shallower depth.

🧪 CLAT-style practice (not a past question) Perimeter: cannot be determined

A rectangular advertising hoarding is to be enlarged: its length by 20% and its breadth by 10%. An LED strip runs along its entire border, and vinyl covers its face. No dimensions of the hoarding are given.

By what percent will the length of LED strip needed increase?

A15%

B30%

C32%

DCannot be determined

Reveal answer + reasoning

Answer: D

The strip follows the perimeter, 2(l + b), which rises by (20l + 10b)/(l + b) %. That depends on the board's shape: a square board gives 15%, one twice as long as it is high gives 16.67%. Without the sides it cannot be determined. The vinyl (area) is fixed at +32%. About 40 seconds.

Why the others fail

A — True only if length equals breadth, which the passage never says.
B — Simply adds 20 + 10.
C — That is the area change, the vinyl, not the strip.

Which scaling myths should you drop?

💥 Myth vs fact

Four scaling myths

🙅 Myth

Every side +10% means area +10%.

Tap to bust it →

✅ Fact

1.1² = 1.21: area +21%, volume +33.1%.

🙅 Myth

Length +20% and breadth −20% leaves the area unchanged.

Tap to bust it →

✅ Fact

1.2 × 0.8 = 0.96: a 4% fall.

🙅 Myth

Doubling a cylinder's radius doubles its volume.

Tap to bust it →

✅ Fact

r is squared: volume × 4. Doubling the height only doubles it.

🙅 Myth

Area up 50% means each side up 25%.

Tap to bust it →

✅ Fact

Take the square root: √1.5 ≈ 1.225, so sides up about 22.5%.

Can you crack these five scaling questions?

🎯 Quick quiz

Five stretches, five answers

0 / 5

Each side of a square photo frame is increased by 30%.

Q1. By what percent does the frame's area increase?

One big round laddoo of radius 6 cm is re-rolled into small laddoos of radius 2 cm, with no dough wasted.

Q2. How many small laddoos are made?

A rectangular notice board's length is cut by 20% and its breadth raised by 25%.

Q3. What happens to its area?

A circular flowerbed is redesigned so that its area falls by 36%.

Q4. By what percent was its radius reduced?

A cube-shaped storage box is replaced by one whose edge is half as long.

Q5. By what percent does the storage volume fall?

🎯 Scaling in a nutshell

Every length × k: perimeter × k, area × k², volume × k³.
Radius, diagonal and slant height are lengths; surface area is an area.
Unequal changes: multiply the factors, raising each to its power in the formula.
Backwards: square root for area, cube root for volume.
Perimeter from unequal percentages alone: cannot be determined.

📎 Formula card

Length × k: perimeter × k; area × k²; volume × k³.
Two sides a% and b%: area change = a + b + ab/100 (minus for a fall). Same x% on both: 2x + x²/100.
Ready numbers: +10% → area +21%, volume +33.1%; +20% → +44%, +72.8%; −10% → −19%, −27.1%.
Cylinder / cone: V ∝ r²h. Sphere: V ∝ r³, surface ∝ r².
Keep area fixed: length +x% → breadth −x/(100 + x) × 100% (+25% → −20%).
Reverse: area × 2 → side × √2 ≈ 1.414; volume × 8 → edge × 2.

Frequently asked questions

How do you calculate the percentage change in area?

Turn each side's change into a multiplier and multiply them. Length +20% and breadth −10% gives 1.2 × 0.9 = 1.08, an 8% rise. In percent form the rule is a + b + ab/100. If both sides change by the same x%, the area changes by 2x + x²/100, so +10% on each side means +21%.

If the side of a square is doubled, what happens to its area?

The area becomes four times as large, because area is side × side and both sides double. A 3 cm square has area 9 cm²; a 6 cm square has 36 cm². The perimeter only doubles, from 12 cm to 24 cm. For a cube, doubling the edge makes the volume eight times larger.

What happens to the volume of a cylinder if its radius is doubled?

It becomes four times larger, because the volume πr²h uses the radius twice. Doubling the height, by contrast, only doubles the volume. If both the radius and the height are doubled, every length has doubled, so the volume becomes 2³ = 8 times as large.

If the area of a square is doubled, by what percent does the side increase?

Take the square root of the area factor. Area × 2 means side × √2, about 1.414, so the side grows by roughly 41.4%, not 100% or 50%. In the same way, an area 44% larger means sides 20% longer, since √1.44 = 1.2.

Can you find the change in perimeter from the percentage changes in the sides?

Only if every side changes by the same percent, in which case the perimeter changes by that percent too. If the length and breadth change by different percents, the perimeter change depends on the actual lengths, so without them the answer cannot be determined. Area does not have this problem.