A cone is one third of its matching cylinder

A cone with the same radius and vertical height as a cylinder fills exactly one third of that cylinder's volume. Use radius and vertical height for volume; use slant height for the tilted side surface.

Calculate cone volume

Identify the radius.
Use vertical height, not slant height.
Calculate circular base area.
Divide the matching cylinder volume by 3.
Keep final units cubic.

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Cone volume and surface relationships

Surface area needs slant height; volume needs vertical height.

Base area

A_{base} = π r^{2}

Matching cylinder

V_{cylinder} = π r^{2} h

Cone volume

V_{cone} = \frac{π r^{2} h}{3} = \frac{V_{cylinder}}{3}

Slant height

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

Total surface area

total surface area = π r^{2} + π r l = π r (r + l)

Diameter conversion

if diameter is given: r = \frac{d}{2}

Common traps

Using diameter as radius.
Using slant height as vertical height for volume.
Mixing units.
Writing volume in square units.
Using surface area when capacity is needed.

FAQ

Why is cone volume divided by 3?

A cone with the same circular base and vertical height as a cylinder fills one third of that cylinder's volume.

Can I use slant height instead of height for volume?

No. Volume uses the perpendicular height from base center to apex. Slant height is for the side surface; if needed, find vertical height with h = sqrt(l^2 - r^2).