Common volumes: the cheat sheet (cube, box, cylinder, sphere, pyramid, cone)

The cheat sheet

Solid	Volume	Note
Cube (side $a$)	$V = a^3$	Simple
Rectangular box ($L, w, h$)	$V = L \times w \times h$	Base area × height
Cylinder (radius $r$, height $h$)	$V = \pi r^2 \times h$	Base area × height
Sphere (radius $r$)	$V = \dfrac{4}{3} \pi r^3$	Only one with $r^3$
Pyramid (base $B$, height $h$)	$V = \dfrac{B \times h}{3}$	One-third of a prism
Cone (radius $r$, height $h$)	$V = \dfrac{\pi r^2 \times h}{3}$	One-third of a cylinder

Two rules to avoid mixing up

Rule 1 (prism/cylinder): $V = \text{base area} \times \text{height}$. Rule 2 (pyramid/cone): $V = \dfrac{\text{base area} \times \text{height}}{3}$.

A cone is a cylinder divided by 3. A pyramid is a box divided by 3.

Watch diameter vs radius

If you’re given the diameter $d$, then $r = d/2$. Many mistakes come from using $r$ instead of $d$ or vice versa.

Volume of a sphere with diameter 6: $r = 3$, $V = \dfrac{4}{3}\pi \times 27 = 36\pi \approx 113$.

Height must be perpendicular

For a cone or pyramid, “height” is the perpendicular distance from base to apex, NOT the slant height.

Units

• Lengths in cm → volume in cm³
• Lengths in m → volume in m³
• Conversion: 1 m³ = 1,000,000 cm³ = 1000 L

?À toi de jouer

What is the exact volume of a sphere with radius 3 cm?

A.36π cm³B.12π cm³C.27π cm³D.108π cm³

Valider