Cone Calculator
Calculate the volume, slant height, lateral surface area, and total surface area of a right circular cone from its base radius and height.
How to use this calculator
- 1Enter the radius of the cone's circular base.
- 2Enter the cone's vertical height from base to tip.
- 3Read the volume and the slant height.
- 4See the lateral, base, and total surface areas.
How it works
Right circular cone
slant height ℓ = √(r² + h²) volume = ⅓ π r² h lateral surface area = π r ℓ total surface area = π r² + π r ℓ
A right circular cone is defined by two measurements: the radius of its circular base and its vertical height from base to apex. From these, everything else follows. The slant height — the length of the sloping side from the rim of the base up to the tip — is the hypotenuse of a right triangle whose legs are the radius and the height, so it equals the square root of the sum of their squares. The volume is one-third of pi times the radius squared times the height, exactly a third of the cylinder that would enclose it. The surface area comes in two parts: the flat circular base, pi times the radius squared, and the curved lateral surface, pi times the radius times the slant height. Importantly, the lateral surface depends on the slant height, not the vertical height, because it wraps along the slope. Adding the base and lateral areas gives the total surface area.
Worked example
A cone with a base radius of 3 and a height of 4 has a slant height of √(3² + 4²) = 5. Its volume is ⅓ × π × 9 × 4 = 12π ≈ 37.70, its lateral surface area is π × 3 × 5 = 15π ≈ 47.12, and its total surface area is 15π + 9π = 24π ≈ 75.40.
Cone Calculator: the complete guide
The one-third rule
One of the most memorable facts in solid geometry is that a cone holds exactly one-third the volume of the cylinder that would just contain it — same circular base, same height. Fill a cone-shaped cup with water and pour it into a cylinder of matching base and height, and it takes precisely three cones to fill the cylinder. This one-third relationship is not a coincidence but a deep result, provable with calculus by summing infinitely many thin circular slices, and it holds for any cone regardless of its proportions.
The same one-third factor appears for pyramids relative to their enclosing prisms, revealing a general principle: a solid that tapers linearly to a point encloses a third of the straight-sided solid with the same base and height. This is why the volume formula for a cone carries that ⅓ out front. Knowing it turns the cone volume into something you can almost do in your head — work out the cylinder (base area times height) and take a third. It is a small piece of mathematical elegance that also happens to be genuinely useful whenever you need the capacity of anything cone-shaped.
Slant height and the curved surface
A subtlety that trips people up is which height to use for surface area. A cone actually has two relevant lengths: the vertical height, straight up from the centre of the base to the apex, and the slant height, running up the sloping outer surface from the rim to the tip. The volume uses the vertical height, but the lateral (curved) surface area uses the slant height, because that curved surface follows the slope, not the vertical line through the middle.
The slant height is found with the Pythagorean theorem, since the vertical height, the base radius, and the slant height form a right triangle inside the cone. Once you have it, the lateral surface area has a beautifully simple form — pi times the radius times the slant height — which can be understood by imagining the curved surface unrolled flat into a sector of a circle. Getting the two heights straight is the key to cone problems: use the vertical height for how much the cone holds, and the slant height for how much material covers its sloping side. Mixing them up is the most common error in cone calculations.
Cones in the world
Cones are among the most common shapes in everyday life and engineering, which makes their measurements practically useful. Ice-cream cones, traffic cones, party hats, funnels, and the pointed tips of rockets and pencils are all cones, and questions about their capacity or the material to make them are cone-volume and cone-surface-area problems. A funnel's throughput, the amount of ice cream a cone holds, or the sheet metal needed to form a conical hopper all come straight from these formulas.
Cones also appear at larger and more natural scales. Volcanoes build themselves into rough cones, piles of sand or gravel settle into cones at a characteristic angle, and the conical shape is structurally efficient for spreading a load from a point down to a wide base, which is why it appears in architecture and in the nose cones of vehicles cutting through air or water. The cone is even one of the classical conic sections — slicing a cone at different angles produces circles, ellipses, parabolas, and hyperbolas, curves that describe planetary orbits and the paths of projectiles. From a scoop of ice cream to the orbit of a comet, the humble cone and the geometry that measures it reach surprisingly far.
Frequently asked questions
How do I find the volume of a cone?
Volume = ⅓ × π × radius² × height. It's exactly one-third of a cylinder with the same base and height. A cone with radius 3 and height 4 has a volume of ⅓ × π × 9 × 4 = 12π ≈ 37.70.
What is the slant height of a cone?
The slant height is the distance up the sloping side from the edge of the base to the tip. It's the hypotenuse of a right triangle formed by the radius and height, so slant = √(radius² + height²). For radius 3 and height 4, the slant height is 5.
How do I find the surface area of a cone?
Add the base and the curved (lateral) surface: total = π r² + π r × slant height. The lateral part uses the slant height, not the vertical height. For radius 3 and slant 5, that's 9π + 15π = 24π ≈ 75.40.
Why is a cone one-third of a cylinder?
Because a cone tapers linearly to a point, it encloses exactly one-third the volume of the cylinder with the same base and height — provable by calculus. It takes three cone-fulls to fill the matching cylinder. The same one-third factor applies to pyramids and prisms.