Volume and Surface Area Formulas

📑 On this page

⚡ The short answer

Volume = how much a solid holds (cubic units). Surface area = how much skin it has (square units). Straight solids: volume = base area × height. A cone is one-third of its cylinder. Fill or melt? Volume. Paint or wrap? Surface. The trap: 1 m³ = 1,000 litres, 1 cu ft = 1,728 cu in.

🎬 The 30-second story

The tank that held three litres

“Terrace tank, open top: 2 m × 1.5 m × 1 m. Painting the inside costs ₹40 per m². How much water does it hold?”

🛢️

Rohan

Rohan needs two numbers.

“2 × 1.5 × 1 = 3. So 3 m² of paint and 3 litres of water.”

🧮

Aarav

“Three litres is a water bottle. 3 m³ is 3,000 litres.”

🍼

Zoya

1 m³ = 1,000 litres.

“Paint goes on walls, not air. Floor 3, walls 2 + 2 + 1.5 + 1.5. That's 10 m², ₹400.”

🖌️

Meher

No lid, so no top face.

Tara's maths: Same three numbers, two jobs. Filling counts unit cubes inside: 3,000 litres. Painting counts unit squares on the faces: 10 m². Ask 'fill or cover?' before you multiply.

What is the difference between volume and surface area?

Build a cube from 27 dice, three per edge. Volume asks how many dice are inside: 27. Surface area asks how many little faces you'd paint outside: 9 on each of 6 faces, so 54. One counts cubes, the other counts squares. Hence cm³ versus cm².

⚔️ Side by side

Fill it or cover it?

🧊Volume

space inside, in cubic units

•Fill, pour, hold, melt, recast
•1 L = 1,000 cm³

🎁Surface area

skin outside, in square units

•Paint, wrap, label, line
•Open box? Drop the lid

Test	Volume	Surface area
Rohan's tank	3 m³ = 3,000 L	10 m² (open top)

What are the volume and surface area formulas of 3D shapes?

📐 The figure

The flat-topped family

Cuboid

V = lbh; TSA = 2(lb + bh + hl)

Cube (all sides a): V = a³, TSA = 6a².

Cylinder

V = πr²h; CSA = 2πrh

TSA = 2πr(r + h): side plus two lids.

Figures are not to scale.

📐 The figure

The pointy and the round

Cone

V = ⅓πr²h; CSA = πrl

Slant l = √(r² + h²).

Sphere

V = 4/3 πr³; SA = 4πr²

Hemisphere: V = 2/3 πr³.

Figures are not to scale.

Why is a cone one-third of a cylinder?

Slice a cuboid or cylinder into thin layers and every layer is a copy of the base. Stack them h high: base area × height. Cuboid lb × h, cylinder πr² × h.

🧮 The formula

One rule for every straight solid

⬛

Base area

lb or πr²

📏

Height

same unit

🧊

Volume

cubic units

A cone narrows to a point, so its layers shrink. Pour three cones of water into a cylinder of the same base and height and it's full: ⅓πr²h. A sphere that fits snugly in a cylinder (height 2r) fills two-thirds of it: ⅔ × 2πr³ = 4/3 πr³. Archimedes wanted that on his tomb.

📊 The data

Cone, sphere, cylinder: radius 3 cm, height 6 cm

Cone · ⅓ × π × 9 × 6

18π cm³

Sphere (r 3) · 4/3 × π × 27

36π cm³

Cylinder · π × 9 × 6

54π cm³

When the height is the diameter, cone : sphere : cylinder = 1 : 2 : 3. One picture, three formulas.

💡 Unroll the label

Peel off a can's label: it's a rectangle, h tall and one circumference (2πr) long. Area = 2πrh, the curved surface. Nothing to memorise.

How do you solve volume and surface area questions?

✍️ Worked example Warm-up ⏱ 40 sec

The gift box

Meher's gift box is 30 cm × 20 cm × 10 cm. How much does it hold, and how much wrapping paper covers it exactly?

1
Hold = volume: 30 × 20 × 10 = 6,000 cm³, i.e. 6 litres.
2
Wrap = total surface: faces 600, 200 and 300 cm², each twice.
3
2 × 1,100 = 2,200 cm².

Answer: 6,000 cm³ inside; 2,200 cm² of paper

⚡ The exam-hall shortcut

Add the three different faces once, then double.

✍️ Worked example CLAT level ⏱ 60 sec

Label and coffee on one can

Kabir's cold-coffee can has radius 3.5 cm and height 12 cm. Label area round its side? Millilitres inside? (π = 22/7)

1
Label = curved surface: 2πrh.
2
22/7 × 3.5 = 11, so 2 × 11 × 12 = 264 cm².
3
Coffee = πr²h = 11 × 3.5 × 12 = 462 cm³ = 462 ml.

Answer: Label 264 cm²; 462 ml of coffee

⚡ The exam-hall shortcut

With r = 3.5 or 7, cancel the 7 in 22/7 first.

✍️ Worked example CLAT level ⏱ 75 sec

Canvas for the fest tent

Zoya's fest tent is a cone, radius 6 m, height 8 m. Canvas needed (no floor)? Air inside?

1
Canvas = curved surface, πrl. You need the slant.
2
r, h, l make a right triangle: l = √(36 + 64) = 10 m.
3
Canvas = 60π ≈ 188.6 m². Air = ⅓ × π × 36 × 8 = 96π ≈ 301.7 m³.

Answer: 60π ≈ 188.6 m² of canvas; 96π ≈ 301.7 m³ of air

⚡ The exam-hall shortcut

Volume uses height; surface uses slant. Put 8 in πrl and you're 20% short.

✍️ Worked example Tricky ⏱ 90 sec

One wax ball, how many candles?

Aarav melts a wax ball of radius 6 cm into candles of radius 1 cm, height 6 cm. How many candles? Is surface area kept?

1
Melting keeps volume. Ball = 4/3 × π × 216 = 288π cm³.
2
One candle = π × 1 × 6 = 6π. Candles = 288π ÷ 6π = 48.
3
Ball surface = 4π × 36 = 144π. The candles' sides alone = 48 × 12π = 576π.

Answer: 48 candles; surface area is not conserved

⚡ π cancels

π always cancels in recasting. Strike it out, then divide.

How do you convert units and count fast in the exam?

1
Name the job
Fill, hold, melt: volume. Paint, wrap, label: surface.
2
Convert first, then cube
1 m³ = 10,00,000 cm³ = 1,000 L. 1 cu ft = 1,728 cu in.
3
Count along the edges
Boxes in a room: divide edge by edge, multiply. Small numbers.
4
Keep π till the end
It often cancels. If not, π ≈ 3 kills far-off options.

🎛️ Try it yourself — change any number

Try it: the 2020 CLAT room

Room length (ft) Room breadth (ft) Room height (ft)

Volume

2,592

Total surface area

1,152

Longest diagonal

24.74

V = lbh; TSA = 2(lb + bh + lh)

2,592 cu ft, 1,152 sq ft of surface. Now type 216, 144, 144 (inches): volume jumps 1,728 times, surface only 144 times. Then try Rohan's tank: 2, 1.5, 1.

⚠️ Four walls is not total surface

Four walls = 2h(l + b): 2 × 12 × 30 = 720 sq ft for the CLAT room. The calculator's 1,152 adds floor and ceiling.

How does CLAT hide volume in a caselet?

CLAT 2020 gave a room and four objects with length, breadth and height. Four questions used only footprints. One switched to 3D: fill the room with books. Read each question's verb: cover means area, fill means volume.

📊 The data

Volumes in the 2020 room caselet

Room · 18 × 12 × 12

2,592 cu ft

Bed · 3.5 × 6 × 3

63 cu ft

Table · 3 × 4 × 4

48 cu ft

Chair · 2 × 2.5 × 4

20 cu ft

Furniture totals 131 cu ft, about 5% of the room. A book is 1 ft × ½ ft × ¼ ft: 8 books per cu ft, so 2,592 × 8. No 1,728s.

Which traps cost marks in 3D questions?

✓Uncubed conversion. 1 m³ = 10,00,000 cm³; 1 cu ft = 1,728 cu in.
✓Diameter as radius. Halve before you square.
✓Height for slant. Cone surface needs l.
✓Phantom lids. Open tank: one base. Pipe: none.

🧭 Decision flow

Which surface does the question want?

❓ Is it about filling, holding or melting?

Yes →

Volume. Convert units first, then cube.

No ↓

❓ Only the side? (label, pipe, tent, four walls)

Yes →

Curved / lateral surface: 2πrh, πrl, 2h(l + b).

No ↓

❓ Is one end open?

Yes →

Side + one base.

No →

Total surface: every face.

How did CLAT ask it? The 2020 room and two CLAT-style caselets

📝 Asked in CLAT 2020 · Q144 Fill the room with books

A room having the dimensions as mentioned in the chart below has a table, a bed and a chair. There are three books kept on the table (side by side) and also a lamp having radius, 3.5 inch… Room: 18 feet × 12 feet × 12 feet Bed: 3.5 feet × 6 feet × 3 feet Table: 3 feet × 4 feet × 4 feet Chair: 2 feet × 2.5 feet × 4 feet Books: 12 inch × 6 inch × 3 inch (length × breadth × height)

If the entire room is to be filled by books having same dimensions as lying on the table then how many books are required.

A2,00,736

B20,736

C11,736

D17,522

Reveal answer + reasoning

Answer: B

Fill = volume; the key treats the room as empty. In inches: 216 × 144 × 144. Per edge: 216 ÷ 12 = 18, 144 ÷ 6 = 24, 144 ÷ 3 = 48. Books = 18 × 24 × 48 = 20,736. Faster: 8 books per cu ft, 2,592 × 8. About 40 seconds. Subtracting the furniture gives 19,688, not offered, so the room is empty.

Why the others fail

A — About ten times too many. Volume says near 20,000.
C — Copies the last digits to tempt a quick eye.
D — No route gives this, even with furniture removed.

🧪 CLAT-style practice (not a past question) Water moved between tanks

A society has a cuboidal underground sump, 8 m × 5 m × 4.5 m inside, and a cylindrical overhead tank of radius 3.5 m and height 4 m. The tank's outside curved surface and top will be painted at ₹30 per m²; its base rests on the roof. (π = 22/7.)

The full sump fills the empty tank. By how much does the sump's water level fall?

A0.65 m

B3.85 m

C4.5 m

DCannot be determined

Reveal answer + reasoning

Answer: B

Water moved = 22/7 × 3.5 × 3.5 × 4 = 154 m³. Sump base = 40 m². Fall = 154 ÷ 40 = 3.85 m. About 60 seconds.

Why the others fail

A — The level left: 4.5 − 3.85.
C — The sump holds 180 m³; only 154 m³ moves.
D — Volume moved = base area × fall. All given.

🧪 CLAT-style practice (not a past question) Paint the tank

Same society: cylindrical tank, radius 3.5 m, height 4 m; outside curved surface and top painted at ₹30 per m²; base unpainted. (π = 22/7.)

What will the painting cost?

A₹2,640

B₹3,795

C₹4,950

D₹9,900

Reveal answer + reasoning

Answer: B

Side = 2πrh = 88 m². Top = πr² = 38.5 m². 126.5 × 30 = ₹3,795. About 50 seconds.

Why the others fail

A — Side only.
C — Both lids; the base sits on the roof.
D — Diameter 7 used as radius.

Myths and a quick test

💥 Myth vs fact

3D myths that cost marks

🙅 Myth

Double every edge and the volume doubles.

Tap to bust it →

✅ Fact

It goes up 8 times (2³). The surface goes up 4 times (2²).

🙅 Myth

Melting and recasting keeps the surface area.

Tap to bust it →

✅ Fact

Only volume. The ball had 144π of surface; the candles' sides have 576π.

🙅 Myth

A cone's surface uses its height.

Tap to bust it →

✅ Fact

Surface uses the slant l = √(r² + h²). Volume uses the height.

🙅 Myth

1 m³ = 100 litres.

Tap to bust it →

✅ Fact

1 m³ = 1,000 litres. Rohan's 3 m³ tank holds 3,000.

🎯 Quick quiz

Five solids, five answers

0 / 5

Q1. A cube has a total surface area of 150 cm². What is its volume?

Q2. A closed tin has radius 7 cm and height 8 cm. What is its total surface area? (π = 22/7)

Q3. A tank measures 2 m × 1.2 m × 0.5 m. How many litres does it hold?

Q4. A 3 cm cube is painted on every face and cut into 1 cm cubes. How many small cubes have exactly two painted faces?

Q5. A cone has radius 5 cm and height 12 cm. What is its curved surface area?

🎯 Volume and surface area in a nutshell

Fill = volume; cover = surface.
Straight solids: base area × height.
Cone = ⅓ cylinder; sphere = ⅔ of its cylinder.
Convert first, then cube.
Melting keeps volume, not surface.

📎 Formula card

Cube: V = a³; TSA = 6a²; diagonal = a√3
Cuboid: V = lbh; TSA = 2(lb + bh + hl); walls = 2h(l + b)
Cylinder: V = πr²h; CSA = 2πrh; TSA = 2πr(r + h)
Cone: V = ⅓πr²h; CSA = πrl; l = √(r² + h²)
Sphere: V = 4/3 πr³; SA = 4πr²
Recast pieces = old volume ÷ new piece volume
1 m³ = 1,000 L; 1 cu ft = 1,728 cu in

Frequently asked questions

What is the difference between volume and surface area?

Volume measures the space inside a solid, in cubic units, so it answers fill, hold and melt questions. Surface area measures the outer skin, in square units, so it answers paint, wrap and label questions. A 2 m × 1.5 m × 1 m open tank holds 3 m³, or 3,000 litres, but has only 10 m² of inside surface to paint.

What is the formula for the volume and surface area of a cylinder?

Volume = πr²h. Curved surface = 2πrh, because the side unrolls into a rectangle 2πr long and h tall. Total surface = 2πr(r + h), adding the two round ends. A can of radius 3.5 cm and height 12 cm holds 462 cm³ and needs a 264 cm² label, taking π as 22/7.

How many litres are there in a cubic metre?

One cubic metre holds 1,000 litres, and one litre is 1,000 cubic centimetres. So convert the volume to m³ and multiply by 1,000. A tank 2 m by 1.2 m by 0.5 m is 1.2 m³, which is 1,200 litres. Don't multiply by 100: the factor is cubed, not squared.

What stays the same when a solid is melted and recast?

Only the volume. Number of new pieces = old volume ÷ volume of one new piece, and π usually cancels. A wax ball of radius 6 cm (288π cm³) makes 48 candles of radius 1 cm and height 6 cm (6π cm³ each). Surface area changes: the candles' sides alone are four times the ball's surface.