EQN-04-05

Solids — Volume & Surface Area

Volume and surface-area formulas for the boxes, cylinders, and cubes the GAT-E uses.

~6 min read•Difficulty 2/5

The formulas you need

Cube (side s) : volume = s^{3} surface area = 6 s^{2}

Box / cuboid (l, w, h) : volume = l \cdot w \cdot h surface area = 2 (lw + lh + wh)

Cylinder (r, h) : volume = π r^{2} \cdot h surface area = 2 π r^{2} + 2 π r \cdot h

Volume is "area of the base $\times$ height"

For any prism or cylinder, volume $= ($ area of the base $) \times$ height. Recognising this means you only memorize base areas, not separate volume rules.

Surface area $=$ add up the faces

Unfold the solid in your mind: a box has 3 pairs of equal rectangular faces; a cylinder has two circles plus one rolled-up rectangle (height $\times$ circumference).

Units and scaling

Volume is cubic units, surface area is square units — keep them separate. And if every length scales by k, surface area scales by k² while volume scales by k³.

Diagram

Volume of a box $=$ length $\times$ width $\times$ height $= 5 \times 3 \times 4.$

Worked examples

Example 1 (difficulty 1/5)

A box measures

5 \times 3 \times 4.

What is its volume?

◎ Predict the answer first:

Example 2 (difficulty 2/5)

If the side of a cube is doubled, the volume becomes how many times larger?

◎ Predict the answer first:

Tips

For any prism or cylinder, volume $=$ base area $\times$ height — one idea covers many shapes.
Keep cubic units (volume) and square units (surface area) clearly apart.
Scaling: lengths $\times k \to$ area $\times k^{2} \to$ volume $\times k^{3} .$

Common mistakes

Doubling a dimension and expecting the volume to double

Why: Linear intuition applied to a 3-D quantity.

Fix: Doubling one edge of a cube multiplies volume by 8, not 2.

Confusing the cylinder volume

π r^{2} h

with surface area

Why: Both use

π, r,

and h.

Fix: Volume fills the inside

(π r^{2} h)

; surface area covers the outside

(2 π r^{2} + 2 π

rh).

← Back to Geometry Practice this lesson →