Learn: Surface Area & Volume

A quick, calm intro — then you choose when to climb.

The idea

Surface area is the wrapping paper — every outside face, in square units. Volume is the filling — everything inside, in cubic units. A cylinder is two circles plus one curved rectangle: SA = 2πr² + 2πrh, and V = πr²h — the base circle stacked up.

Surface Area = wrap

Outside paper covering the shape (cm²)

Volume = fill

Inside space the shape holds (cm³)

Surface Area

unit: cm²

A cylinder unwraps into 2 circles + 1 rectangle.

Cylinder

Unwrap (net)

Build the formula

1) Two ends:

2\pi r^{2}

2) Side rectangle:

(2\pi r)\cdot h

SA = 2\pi r^{2} + 2\pi r h

Volume

unit: cm³

Volume is the base circle stacked up to height $h$ .

Build the formula

1) Base area:

\pi r^{2}

2) Times height:

\times h

V = \pi r^{2} h

Memory check

Wrap paper → Surface Area →

cm^{2}

Filling inside → Volume →

cm^{3}

Surface Area = wrap

All outside faces added together (cm²)

Volume = fill

How much space is inside (cm³)

Cuboid (Box)

SA:

SA = 2(lw + lh + wh)

V = lwh

Cube

SA:

SA = 6s^{2}

V = s^{3}

Cylinder

SA:

SA = 2\pi r^{2} + 2\pi r h

V = \pi r^{2} h

Cone

SA:

SA = \pi r^{2} + \pi r l

V = \tfrac{1}{3}\pi r^{2} h

l = slant height

Sphere

SA:

SA = 4\pi r^{2}

V = \tfrac{4}{3}\pi r^{3}

Quick memory

If the question says wrap / paint / cover → Surface Area ( $cm^{2}$ )
If the question says fill / capacity / how much inside → Volume ( $cm^{3}$ )
For cylinder/cone/sphere, check whether you were given r or diameter first.

Worked example

Every step explained — why we do it, and what happens.

1
$V = \pi r^2 h = \frac{22}{7} \times 7^2 \times 10$
Why: The base is a circle of area πr²; stacking h copies of it fills the cylinder.
What happens: Volume = base circle × height.
2
$= 154 \times 10 = 1540 \text{ cm}^3$
Why: (22/7) × 49 = 154 — the base circle of a radius-7 cylinder.
What happens: V = 1540 cm³ — the filling, in cubes.
3
$SA = 2\pi r^2 + 2\pi r h = 2\pi r (r + h)$
Why: Two circular ends (2πr²) plus the curved side, which unrolls into a rectangle: width 2πr (the rim), height h.
What happens: Surface area = 2πr(r + h) — one tidy formula.
4
$= 44 \times 17 = 748 \text{ cm}^2$
Why: 2πr with r = 7 is 44 — the rim of the base circle; r + h = 17.
What happens: SA = 748 cm² — the paper, in squares. Cubes fill, squares wrap; a wrong unit means a wrong formula.

Real-world example

A bakery sells cakes in cylindrical tins, radius 7 cm and height 10 cm (use π = 22/7). The cake fills the whole tin — how much cake is that? And how much gift wrap covers the whole tin — top, curved side and bottom?

1
$V = \pi r^2 h$
Why: The cake fills the tin — that is volume: base circle × height.
What happens: V = (22/7) × 7² × 10.
2
$= 154 \times 10 = 1540 \text{ cm}^3$
Why: Base area 154 cm² (radius 7), times 10 cm of height.
What happens: 1540 cm³ of cake.
3
$\text{Trap: wrap } = \pi r^2 h = 1540 \text{ cm}^2\,?$
Why: The glitch: using the volume formula for the paper. Gift wrap covers the OUTSIDE — every face — so it must be surface area, in square units.
What happens: 1540 cm² would be the volume in disguise — wrong formula, wrong unit.
4
$SA = 2\pi r (r + h) = 44 \times 17 = 748 \text{ cm}^2$
Why: Two ends (2πr²) plus the curved side (2πrh) — factor 2πr; with r = 7 that is 44, and r + h = 17.
What happens: 748 cm² of gift wrap.
5
$748 \text{ cm}^2 \ne 1540 \text{ cm}^3$
Why: The check: squares versus cubes. 1540 fills the tin, 748 wraps it — different jobs, different formulas, different units.
What happens: V = 1540 cm³, SA = 748 cm² ✓

Common glitch

Two classic glitches:

1.Using the diameter as the radius

Wrong:A cylinder with diameter 14 has r = 14 — the diameter is just a bigger radius

Right:Halve it first: diameter 14 → r = 7. Every formula is built on r, so one wrong r doubles every answer.

2.Volume for the paper, area for the filling

Wrong:πr²h and 2πr² + 2πrh are interchangeable — pick whichever looks nicer

Right:Filling → V = πr²h (cubes). Paper → SA = 2πr² + 2πrh (squares). The unit of your answer tells you which formula you used.

Volume and Surface Area Calculator — Calculate volume and surface area for common solids.
Unit Converter — Convert dimensions to consistent units before calculating.

Next up: Probability →

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Common glitch

Two classic glitches:

1.Using the diameter as the radius

Wrong:A cylinder with diameter 14 has r = 14 — the diameter is just a bigger radius

Right:Halve it first: diameter 14 → r = 7. Every formula is built on r, so one wrong r doubles every answer.

2.Volume for the paper, area for the filling

Wrong:πr²h and 2πr² + 2πrh are interchangeable — pick whichever looks nicer

Right:Filling → V = πr²h (cubes). Paper → SA = 2πr² + 2πrh (squares). The unit of your answer tells you which formula you used.

Learn: Surface Area & Volume

The idea

Surface Area

Volume

Worked example

Real-world example

Common glitch

Try a related calculator

Learn: Surface Area & Volume

The idea

Surface Area

Volume

Worked example

Real-world example

Common glitch

Try a related calculator