Volumes of cone, pyramid, and sphere: the 3 formulas

The 3 formulas to memorize

Solid	Volume
Right circular cone	$V = \dfrac{\pi r^2 h}{3}$
Pyramid (any base)	$V = \dfrac{B \times h}{3}$ where $B$ = base area
Sphere	$V = \dfrac{4}{3} \pi r^3$

Memory hint: cone and pyramid are one third of the prism/cylinder with the same base and height.

Example: cone

Radius $r = 3$ cm, height $h = 6$ cm.

$V = \dfrac{\pi \times 3^2 \times 6}{3} = \dfrac{54 \pi}{3} = 18 \pi \approx 56.5 \text{ cm}^3$.

Example: square-base pyramid

Base side $c = 4$ cm, height $h = 9$ cm.

$B = 4^2 = 16 \text{ cm}^2$, $V = \dfrac{16 \times 9}{3} = 48 \text{ cm}^3$.

Example: sphere

Radius $r = 5$ cm.

$V = \dfrac{4}{3} \pi \times 5^3 = \dfrac{500 \pi}{3} \approx 523.6 \text{ cm}^3$.

Bonus: sphere surface area

$\mathcal{A}_{\text{sphere}} = 4 \pi r^2$

Also 4 times the area of the disk with the same radius.

Units

• Lengths in cm ⇒ Volumes in cm³
• Lengths in m ⇒ Volumes in m³

Conversions: $1 \text{ dm}^3 = 1 \text{ L}$, $1 \text{ m}^3 = 1000 \text{ L}$.

Common mistakes

1. Forgetting the $/3$ for cone or pyramid.
2. $V_{\text{sphere}} = \pi r^3$ (wrong, it’s $\dfrac{4}{3} \pi r^3$).
3. Confusing radius and diameter (r = d/2).
4. Wrong volume unit (missing the cube).

?À toi de jouer

Volume of a sphere with radius 3 cm (π ≈ 3.14)?

A.≈ 113 cm³B.≈ 28 cm³C.≈ 36 cm³D.≈ 85 cm³

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