Surface area and volume of all above 3D shapes: Question bank

Before we start, one word we lean on constantly:

True or false — justify

TF1. A cube and a sphere with the same surface area have the same volume.

False. For a fixed surface area the sphere always encloses the most volume of any shape — that's why bubbles are round. The cube wastes "skin" on corners, so it holds less.

TF2. Doubling the radius of a sphere doubles its volume.

False. Volume is

\frac{4}{3} π r^{3}

, so it scales with

r^{3}

; doubling

r

multiplies volume by

2^{3} = 8

, not

2

. See Similar Solids.

TF3. Doubling every edge of a cube doubles its surface area.

False. Surface area scales with (length)

^{2}

, so it multiplies by

2^{2} = 4

. Only a length (like the perimeter of one face) would double.

TF4. A cone and a cylinder with the same base and same height have the same volume.

False. The cone is exactly one-third of that cylinder (

\frac{1}{3} π r^{2} h

π r^{2} h

) because its cross-sections shrink to a point as you climb.

TF5. The slant height

l

of a cone is always longer than its vertical height

h

True. Since

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

and

r > 0

, we add a positive amount under the root, so

l > h

always — see Pythagorean Theorem.

TF6. A hemisphere's total surface area is exactly half a sphere's surface area.

False. The curved part is half (

2 π r^{2}

), but the hemisphere also gains a flat circular lid (

π r^{2}

), giving

3 π r^{2}

— more than half of

4 π r^{2}

TF7. Two boxes with the same volume must have the same surface area.

False. A

1 \times 1 \times 8

box and a

2 \times 2 \times 2

box both have volume

8

, but the long thin one has far more surface area. Volume fixes the "stuffing," not the "wrapping."

TF8. If you know the surface area of a cube you can find its volume.

True.

S A = 6 a^{2}

gives

a

uniquely (edge is positive), and

a

then gives

V = a^{3}

. One number pins the whole cube because it has only one free length.

TF9. The formula

V = π r^{2} h

works for a tilted (oblique) cylinder too.

True. By Cavalieri's principle, sliding the slices sideways doesn't change any slice's area, so the volume is unchanged as long as base area and vertical height stay the same.

Why questions

WHY1. Why is a cone's volume one-third of the cylinder's, not one-half?

Because the cross-sectional circle shrinks with the square of how far up you are, and averaging

x^{2}

over the height gives

\frac{1}{3}

, not

\frac{1}{2}

. Integration confirms the exact third.

WHY2. Why does the curved surface of a cylinder unroll into a perfect rectangle?

Because at every height the boundary is the same circle, so the wrap has constant width equal to the circumference

2 π r

; a shape of constant width and height

h

is a rectangle of area

2 π r h

WHY3. Why does the cone's curved surface unroll into a sector (pizza slice), not a rectangle?

The wrap's width shrinks to zero at the tip, so it fans out from a point — a sector of radius

l

whose arc length is the base circumference

2 π r

WHY4. Why is slant height, not vertical height, the "radius" of that unrolled sector?

Every point on the base rim sits a straight-line distance

l

from the apex, and unrolling preserves that distance, so all rim points land at radius

l

WHY5. Why do we use the Pythagorean theorem to find slant height at all?

Because

r

h

, and

l

form a right triangle (the axis is vertical, the radius horizontal, meeting at

9 0^{\circ}

), and Pythagoras is exactly the tool for the third side of a right triangle — see Pythagorean Theorem.

WHY6. Why does surface area use squared units and volume cubed units?

Area multiplies two lengths (length² ), volume three (length³). Tracking these units catches formula errors instantly — a "volume" that came out in cm² must be wrong (see Dimensional Analysis).

WHY7. Why can two solids have equal volume but different mass?

Mass depends on material packed into the volume:

mass = density \times volume

. Lead and foam of equal size differ wildly — see Density and Mass.

WHY8. Why does the sphere's surface area (

4 π r^{2}

) match a cylinder's side that hugs it?

Archimedes showed that as you slice both, the sphere's slices flatten exactly as much as they widen, so each ring's area equals the enclosing cylinder ring's — the two lateral areas coincide at

4 π r^{2}

Edge cases

EC1. What is the volume of a cone when its height

h = 0

Zero.

V = \frac{1}{3} π r^{2} \cdot 0 = 0

— a "cone" of no height is just a flat disc, which encloses no space.

EC2. What happens to

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

for a cone as

r \to 0

l \to h

. With no base radius the cone collapses to a vertical needle, so slant and height coincide.

EC3. For a cone with

h = 0

but

r > 0

, what is its total surface area?

l = r

, so

T S A = π r (r + r) = 2 π r^{2}

— the flat disc gets counted on both faces, base and "slant" now flattened onto it.

EC4. If a cuboid has

l = w = h = a

, do its formulas become the cube's?

Yes —

V = a \cdot a \cdot a = a^{3}

and

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

. A cube is just a cuboid with all edges equal, so the general formulas must specialize correctly.

EC5. What is the surface area of a sphere with

r = 0

Zero.

S A = 4 π (0)^{2} = 0

— a point has no surface, and the formula honours that limiting case.

EC6. A hemisphere shrinks to

r = 0

; what happens to its curved-to-flat area ratio?

The ratio stays

2 π r^{2} : π r^{2} = 2 : 1

for all

r > 0

, but both areas go to zero as

r \to 0

; the shape proportion is scale-free even as size vanishes — see Similar Solids.

EC7. Can a pyramid's slant height ever equal its vertical height?

Only in the degenerate limit

a \to 0

(a needle). For any real square base

a > 0

l = h^{2} + (a /2)^{2} > h

EC8. If you double the radius and halve the height of a cylinder, does volume stay the same?

No —

V = π r^{2} h

, so doubling

r

multiplies by

4

, halving

h

divides by

2

, net factor

2

. Volume doubles because the radius is squared but height is not.

Recall Self-test before you close

Which single quantity, known alone, is enough to reconstruct an entire cube? ::: Any one of edge $a$ , surface area, or volume — each pins down $a$ , and $a$ determines everything. Why does scaling a solid by factor $k$ multiply its volume by $k^{3}$ but surface area by $k^{2}$ ? ::: Volume is a product of three lengths and area of two; each length scales by $k$ , so the powers follow (see Similar Solids).

Question bank — Surface area and volume of all above 3D shapes

True or false — justify

Spot the error

Why questions

Edge cases