3D shapes — cube, cuboid, cylinder, cone, sphere, prism, pyramid: Question bank

Definition Symbols and shorthand used on this page

Before we use any symbol, here is what each one means — anchored to a picture you can hold in your head:

$a$ — the edge length of a cube: the length of any one of its 12 identical straight sides (they're all equal, so one letter $a$ covers them all).
$l, w, h$ — the length, width, height of a box (cuboid), the three different straight measurements meeting at a corner.
$r$ — the radius of a circle (the straight distance from centre to rim), used for cylinders, cones and spheres.
$h$ (round shapes) — the perpendicular height: the straight up-and-down distance between base and top/apex, measured at a right angle to the base — not along a slanted edge.
$l$ (cone) — the slant height: the straight distance along the tilted surface from the rim of the base up to the apex.
SA — just shorthand for surface area (total area of the outside skin). We write "SA" to save space; it always means square units.
$A_{base}$ — the area of the base (the flat 2D shape you'd stack).
$P_{base}$ — the perimeter of the base: the total distance once around the edge of that base shape (for a circle this is the circumference $2 π r$ ).

True or false — justify

Doubling every edge of a cube doubles its volume.

False. Volume is

a^{3}

(with

a

the edge length), so doubling

a

multiplies volume by

2^{3} = 8

. A twice-as-tall, twice-as-wide, twice-as-deep box holds eight of the originals.

Doubling every edge of a cube doubles its surface area.

False. Surface area is

6 a^{2}

, so doubling

a

multiplies it by

2^{2} = 4

. Area scales with the square, volume with the cube — this gap is the whole story of why big animals are bulky.

A cylinder and a cone with the same radius and height have the same volume.

False. The cone holds exactly one-third:

V_{cone} = \frac{1}{3} π r^{2} h

versus

V_{cyl} = π r^{2} h

. Three identical cones of water fill the cylinder.

The slant height

l

of a cone is always longer than its vertical height

h

True (unless

r = 0

). Since

l = r^{2} + h^{2}

and

r^{2} \geq 0

, we always get

l \geq h

, with equality only when the radius is zero (a degenerate needle).

For a sphere, surface area equals four times the area of the flat circle you'd get by slicing through its centre.

True. The great circle has area

π r^{2}

; the sphere's surface is

4 π r^{2} = 4 \times (π r^{2})

— a genuinely surprising exact factor, first proved by Archimedes.

A prism and a cylinder are unrelated shapes.

False. A cylinder is the limiting case of a prism whose base is a polygon with more and more sides — as the sides grow without bound, the polygon becomes a circle. Since every prism has volume

(base area) \times h

no matter how many sides its base has, the limit inherits the same rule, giving

V = π r^{2} h

If two cuboids have equal volume, they must have equal surface area.

False. A

1 \times 1 \times 8

box and a

2 \times 2 \times 2

cube both have volume

8

, but surface areas

34

and

24

respectively. Long thin shapes waste more skin per unit of volume.

A cube is a special kind of cuboid, and also a special kind of prism.

True. A cube is a cuboid with

l = w = h

, and a prism whose base is a square — every property of the general shapes must still hold for it.

Surface area and volume of the same shape can be numerically equal.

True (as numbers, not as quantities). E.g. a cube with

a = 6

gives

S A = 216

and

V = 216

. But the units differ (cm² vs cm³), so it's a coincidence of numbers, not a physical equality.

The curved surface of a cone, unrolled, is a full flat circle.

False. It unrolls into a sector (a pie-slice) of a circle of radius

l

, whose arc length equals the base circumference

2 π r

— never the whole disc unless

r = l

, which can't happen for a real cone (see the unrolling figure below).

Why questions

Why is a cone exactly one-third of the matching cylinder, not one-half?

As you rise from base to apex, the circular slices shrink; their areas fall off as the square of the shrinking radius, so on average a slice is one-third the base — this is what the integral of

r^{2}

delivers, giving the clean

\frac{1}{3}

(see the slicing figure below).

Why does the curved surface of a cylinder unroll into a rectangle?

The wall is straight up-and-down (no taper), so slicing it vertically and flattening gives a rectangle of width

2 π r

(the base circumference) and height

h

— hence its area is

2 π r h

Why does the cone's curved surface unroll into a sector instead of a rectangle?

The cone's wall does taper toward the apex, so straight vertical cuts flattened out fan open into a pie-slice of radius

l

whose curved edge is the base circumference

2 π r

; its area works out to

π r l

Figure — 3D shapes — cube, cuboid, cylinder, cone, sphere, prism, pyramid

Why does area scale as length² but volume as length³?

Area covers a 2-directional patch, so scaling by

k

stretches it in 2 directions (

k^{2}

); volume fills a 3-directional region, stretched in 3 directions (

k^{3}

). This is the deep reason ants can lift many times their weight but elephants cannot. See Real-World Applications.

Why do the sphere and cone volume derivations both need integration but the cube doesn't?

A cube's cross-sections are all identical, so stacking is just multiplication. A sphere and cone have changing cross-sections, so we must sum infinitely many shrinking slices — exactly what Integration does (see the slicing figure).

Why does the sphere's surface area carry the exact factor

4 π r^{2}

— four great circles?

Archimedes' insight is that the sphere's surface has the same area as the curved wall of the cylinder that just encloses it (radius

r

, height

2 r

): that wall is

2 π r \times 2 r = 4 π r^{2}

. Any horizontal band on the sphere is squashed sideways but stretched vertically by exactly the matching amount, so its area is preserved.

Why is

π

in every round-shape formula but never in the cube or cuboid ones?

π

enters only through circles and their areas and circumferences. Cubes and cuboids are built from flat rectangles with no curves, so no

π

can appear.

Why do we add the two bases for a cylinder but the cone only has one base?

A cylinder is capped at both ends (two circles), while a cone tapers to a single point (apex) with no top circle — so its surface is one base plus the curved sheet.

Spot the error

"Surface area of a cylinder is

2 π r^{2} h

." — find the mistake.

The two bases give

2 π r^{2}

and the curved wall gives

2 π r h

; these are added, not multiplied. Correct:

S A = 2 π r^{2} + 2 π r h = 2 π r (r + h)

"Volume of a cone with

r = 3, h = 4

\frac{1}{3} π (3) (4) = 4 π

." — find the mistake.

The base area is

π r^{2} = π (3^{2})

, not

π r

. Correct:

V = \frac{1}{3} π (9) (4) = 12 π

. The radius must be squared for a base area.

"For a cone I plugged the slant height into the volume formula." — why is that wrong?

Volume uses the perpendicular height

h

(base-to-apex), because volume stacks flat slices vertically. The slant height

l

only belongs in the curved surface area

π r l

"Cuboid surface area is

l w + l h + w h

." — find the mistake.

That counts only one of each pair of opposite faces. Every face has a twin, so multiply by 2:

S A = 2 (l w + l h + w h)

"Since a sphere is round, its volume is

\frac{4}{3} π r^{2}

." — find the mistake.

Volume is a 3D quantity and must carry

r^{3}

V = \frac{4}{3} π r^{3}

. The

r^{2}

version is the surface area (times 4), not the volume.

"I found the cone's slant height as

l = r + h = 3 + 4 = 7

." — why is that wrong?

r

and

h

meet at a right angle, so they are the two legs of a right triangle, not addable straight. Use Pythagorean Theorem:

l = r^{2} + h^{2} = 9 + 16 = 5

"A prism's lateral (side) area is (base area)

\times h

." — find the mistake.

The side walls wrap around the perimeter

P_{base}

, not the area. Lateral area

= P_{base} \times h

; only the total volume uses the base area times

h

"Volume came out in cm² so I wrote cm²." — why is that a red flag?

Volume is always cubic (cm³). Getting square units means you multiplied only two lengths and forgot the third dimension — go back and check. See Units and Measurement.

Edge cases

What is the volume of a cylinder when the height

h = 0

Zero. With no height there's nothing to stack — the shape collapses to a flat disc, which has area but no volume.

What happens to a cone's shape and volume as the radius

r \to 0

It degenerates into a vertical line segment of length

h

: volume

\frac{1}{3} π r^{2} h \to 0

, and the slant height

l = r^{2} + h^{2} \to h

. It becomes an infinitely thin needle.

What happens to a cone as the height

h \to 0

but

r

stays fixed?

It flattens into the base disc: volume

\to 0

, slant height

l \to r

, and the curved surface

π r l \to π r^{2}

, matching the flat circle it collapses onto.

Is a cylinder still valid if radius and height are equal (

r = h

Yes — nothing in the formulas requires

r \neq = h

. You simply get

V = π r^{3}

and

S A = 4 π r^{2}

; it's a perfectly ordinary "square-profile" can.

Can a prism have a triangular base, and does the formula still hold?

Yes.

V = A_{base} \times h

works for any polygon base; for a triangle you just use

A_{base} = \frac{1}{2} (base) (height)

of that triangle. The prism formula never assumes a rectangle.

What does

S A = 4 π r^{2}

give when

r = 0

Zero — a sphere of no radius is a single point with no surface. Both volume and area vanish, which is the sensible limiting behaviour.

Are negative dimensions like

r = - 3

h = - 2

allowed?

No — every length in these formulas must satisfy

r \geq 0

h \geq 0

a \geq 0

l, w \geq 0

. A negative length has no physical meaning; even though

a^{3}

π r^{2} h

would still compute a number, it describes no real solid, so such inputs are outside the domain.

Why must we insist on the domain

r \geq 0

and

h \geq 0

rather than just "plugging in whatever"?

The formulas were derived by measuring real distances (stacking slices, unrolling walls), and distances are never negative. Feeding in a negative value can even flip a sign — e.g.

\frac{1}{3} π r^{2} h

with

h < 0

gives a "negative volume", which is nonsense; the guard

r, h \geq 0

keeps every answer physically meaningful.

If a cuboid has

l = w = h

, do the cuboid formulas still work?

Yes, and they must collapse to the cube's:

V = l \cdot w \cdot h = a^{3}

and

S A = 2 (l w + l h + w h) = 6 a^{2}

. A correct general formula always reproduces the special case.

Recall Quick self-check before you leave

"SA" is shorthand for what? ::: Surface area — the total area of a shape's outside skin, in square units. Which quantity scales with the cube of size? ::: Volume (length³); surface area scales only with the square. Where does slant height belong — volume or surface area of a cone? ::: Surface area only ( $π r l$ ); volume uses perpendicular height $h$ . Why is a cone one-third of its cylinder? ::: Slices shrink as radius², averaging to one-third of the base. What does $P_{base}$ mean? ::: The perimeter (distance once around the edge) of the base shape.