The cone is everywhere -- ice cream cones, party hats, traffic markers, funnels and the conical roofs of historic buildings.
About this calculator
The cone is everywhere -- ice cream cones, party hats, traffic markers, funnels and the conical roofs of historic buildings. Mathematically, a right circular cone is defined by just two numbers: the radius of its circular base and its perpendicular height. From these, every other property (slant height, surface area, volume) follows from the same simple formulas first derived by Archimedes.
The most counter-intuitive fact about cones is the (1/3) factor in the volume formula. A cone has exactly one-third the volume of a cylinder with the same base radius and height. This is true for any cone -- a fact you can verify by water-pouring, but Archimedes proved it rigorously over 2200 years ago using the method of exhaustion (a precursor to integral calculus). The same (1/3) factor appears in the volume of any pyramid, generalising to all 'cone-like' solids with a vertex above a flat base.
Real-world cone problems often involve unrolling the lateral surface into a flat sector of a circle. The radius of that sector equals the slant height, and the arc length equals the circumference of the base. This is the basis of cone-making in tin-smithing, paper-folding and even 3D printing. The total surface area combines the lateral surface and the circular base, useful for paint, fabric and material costs. For more 3D shapes use the [[cylinder]] and [[sphere]] calculators -- you'll see how all three are mathematically related (volume of a hemisphere = volume of cylinder − volume of cone with the same base and height). For purely 2D area work use the [[area]] calculator, and for repetitive geometry problems the [[quadratic]] calculator handles the polynomial relationships behind many derivations.
Common uses
- Class 9/10/11 geometry homework -- cone volume and surface area problems
- Calculating paint or material needed to coat conical roofs and structures
- Volume of conical tanks, hoppers and funnels in industry and chemistry
- Designing party hats, ice cream cones and packaging using flat sector layouts
- 3D modelling and CAD where cone parameters need to be derived
Frequently asked questions
What's the difference between height and slant height?
Height is the perpendicular distance from the centre of the base to the apex. Slant height is the distance from any point on the base edge to the apex along the cone
Why is cone volume one-third of cylinder volume?
Imagine inscribing a cone inside a cylinder with the same radius and height. The cone takes up exactly one-third of the cylinder
Does the formula work for an oblique cone?
The volume formula (1/3) × π × r² × h works for any cone where h is measured perpendicular to the base plane, even if the apex is not directly above the centre. However, slant height and lateral area formulas only apply to right circular cones (apex above centre).
How do I find radius if I only know slant height and height?
Use the Pythagorean relation: r = √(ℓ² − h²). For example, slant height 13 and height 12 gives radius √(169 − 144) = √25 = 5.
What units should I use?
Use any consistent linear unit (cm, m, inches, feet) for radius and height. Surface area comes out in square units of the same scale, and volume in cubic units. The L conversion the calculator shows assumes you used cm for input.