How this instrument works
A right circular cone is fully pinned down by just two lengths — the base radius and the perpendicular height from base to apex — yet a complete description of the solid runs to four figures: how far up the slope runs, how much it holds, how much material wraps the sloped side, and how much material covers the whole thing including the bottom. Handled one at a time, that means computing the slant height by hand first, then feeding it into a separate area formula, then starting over with a different formula for volume. This instrument runs the whole chain at once: it derives the slant height from radius and height, then reuses that single value for both surface figures while volume takes its own, entirely separate route through the same two inputs.
The two area outputs and the volume output do not actually depend on the same inputs in the same way, which is easy to miss when all four numbers appear together. Both A_L = πrl and A = A_L + πr² lean on the slant height l, so a typo in the height field quietly shifts every area on the page through that shared intermediate value. Volume, by contrast, is V = ⅓πr²h — it never touches l at all, and depends on the raw height directly. Push h up while holding r fixed and volume climbs in step with it, but the slant height, and both areas built from it, only creep upward through the square root of h², growing far more slowly than the volume beside them.
Scale an entire cone up uniformly — double every length, radius and height together — and the four outputs pull apart from each other. Slant height doubles, since it is a length. Both areas quadruple, since each is built from a product of two lengths. Volume, built from three lengths multiplied together, grows eightfold. That mismatch is why a large storage cone is proportionally cheaper to skin in metal than a small one holding the same shape: the material scales with the square of the size while the capacity scales with the cube, so bigger vessels get more storage for every unit of surface they need.
- Enter the cone's Radius, measured from the centre of the circular base out to its rim.
- Enter the Height, the perpendicular distance from that centre straight up to the apex — not the length along the sloped side.
- Read Slant height first; it is the hypotenuse this sheet builds from Radius and Height before touching either area formula.
- Read Volume for the capacity enclosed, and Lateral area for the curved side alone, wrapping-paper style.
- Read Total surface area last — it is Lateral area plus the flat base, the figure for a cone closed at the bottom.
Worked example — a hopper built at radius 3, height 4
A small conical feed hopper is welded with a base Radius of 3 units and a Height of 4 units. This sheet's first move is the slant height: l = √(3² + 4²) = √25 = 5.0 units exactly, since 3 and 4 happen to be the legs of the smallest whole-number right triangle. That 5.0 becomes the working number for everything below the base measurements.
Volume comes from radius and height directly, skipping the slant height entirely: V = ⅓ × π × 3² × 4 = 12π = 37.69911184307752 cubic units — how much feed the hopper holds. Lateral area reuses the slant height just found: A_L = π × 3 × 5 = 15π = 47.12388980384689 square units of sloped steel. Total surface area adds the base back in: A = 47.12388980384689 + π × 3² = 24π = 75.39822368615503 square units, the full order if the hopper is capped rather than left open at the bottom.
Questions
What does this calculator compute from a radius and height?
Four figures in one pass: the slant height (by the Pythagorean theorem), the volume, the lateral surface area of the sloped side, and the total surface area including the base. Slant height is worked out first because both area formulas need it; volume is worked out separately because it never uses slant height at all.
Does the volume formula use the slant height?
No — V = ⅓πr²h depends only on the base radius and the perpendicular height, never on the slant height l. This surprises people because the slope looks like it should matter for capacity the way it matters for surface area, but a cone's volume is built from horizontal disc slices stacked by height, and slant length plays no part in how much each disc holds.
Why compute slant height before either surface area?
Because both A_L = πrl and A = A_L + πr² are written in terms of l, not the raw height h. Deriving l = √(r² + h²) once and reusing it for both formulas avoids repeating the same square root twice and keeps the two area figures consistent with each other — if one changes, the other moves with it through the shared slant height.
How does surface area scale compared to volume as a cone grows?
Slower. Scale every length of a cone by a factor k and both areas grow by k², while volume — built from three multiplied lengths instead of two — grows by k³. Double a cone's size and its skin only quadruples while its capacity grows eightfold, which is why large conical tanks and hoppers need proportionally less material per unit of storage than small ones of the same shape.
What is the difference between lateral area and total surface area?
Lateral area, A_L = πrl, covers only the sloped side — the number that matters for a hopper, funnel, or party hat left open at the bottom. Total surface area, A = A_L + πr², adds the flat circular base on top, the figure needed when the cone is closed, capped, or sitting solid on a surface.
What happens to these four values as the height shrinks toward zero?
Volume shrinks to zero, since it carries a factor of h directly. The slant height shrinks down until it equals the radius, and both area formulas settle on nonzero values — lateral area approaches πr² and total area approaches 2πr² — because a cone that has gone perfectly flat still has a top face and a bottom face even though it can no longer hold anything.