This calculator computes the full geometry of a right circular cone from just its base radius and height. Enter those two values and it returns the volume, slant height, lateral (curved) surface area, base area and total surface area. It is useful for geometry homework, engineering, and practical jobs like sizing a funnel, hopper or party hat. Keep your units consistent and read all results instantly.
How it works
From the radius (r) and height (h), the tool first finds the slant height using the Pythagorean theorem, then applies the standard cone formulas:
| Quantity | Formula |
|---|---|
| Slant height (l) | √(r² + h²) |
| Volume | ⅓ · π · r² · h |
| Lateral surface area | π · r · l |
| Base area | π · r² |
| Total surface area | π · r · l + π · r² |
It uses full-precision π before rounding for display.
Example
A cone with radius 3 and height 4:
slant = √(3² + 4²) = √25 = 5 volume = ⅓ · π · 9 · 4 = 12π ≈ 37.699 lateral area = π · 3 · 5 = 15π ≈ 47.124 total area = 15π + 9π = 24π ≈ 75.398
So this cone has a volume of about 37.70 and a total surface area of about 75.40 cubic/square units. Everything updates instantly in your browser.
Understanding slant height vs. vertical height
The height of a cone is the perpendicular distance from the center of the base to the apex — the straight vertical measurement. The slant height is the distance along the sloped surface from the apex to any point on the base circle. These are related by the Pythagorean theorem: the slant height is the hypotenuse of the right triangle formed by the vertical height and the base radius.
In practice, slant height matters when you are working with the surface of the cone — for example, calculating how much material is needed to make a conical hat, a funnel, or the sloped side of a hopper. Vertical height matters when you are calculating how much space or volume the cone occupies.
Lateral vs. total surface area
Lateral surface area covers only the curved side — the sloped face of the cone, not the circular base. This is what you need if the cone is open at the bottom (like a funnel or a traffic cone): the formula is π × r × slant height.
Total surface area adds the circular base: π × r². Use this if you are enclosing the cone on all sides — for example, manufacturing a conical hopper that needs material on all faces.
Practical uses
Cone geometry comes up in more places than geometry class:
- Funnels and hoppers: the volume formula tells you the capacity; the lateral surface area tells you how much sheet metal to cut.
- Ice cream cones and party hats: the lateral surface area gives the amount of material needed to wrap the curved surface (a flat cone “net” when unrolled is a sector of a circle).
- Grain and sand piles: material poured onto a flat surface forms a cone; knowing the angle of repose gives you the height-to-radius ratio, and the volume formula gives you the quantity.
- Drainage and landscaping: conical shapes appear in drainage inlets and sloped grade calculations.
Notes on units
The calculator works in any unit as long as you use the same unit for radius and height. Enter radius in centimeters and height in centimeters, and you get volume in cm³ and areas in cm². Convert afterward if needed: 1 cm³ = 0.001 liters; 1 m³ = 1,000 liters.