Cone Volume Calculator
Enter the base radius and height of a cone to get its volume, surface area and slant height, drawn to scale. Click the diameter or height on the drawing to change it.
Dimensions
ConeResult
Volume of a cone
A cone holds exactly one third of the cylinder that would enclose it, so its volume is ⅓πr²h. The slant height — the distance up the sloping side — is found from the radius and height by Pythagoras, and sets the curved lateral area πrl.
Editing on the drawing
Click the base diameter or the height on the figure, enter a value and press Enter to redraw the cone to scale.
How to find the volume of a cone
A cone rises from a circular base to a single point. Two measurements set it: the base radius and the vertical height. A third length, the slant height up the sloping side, is needed for surface area and comes straight from Pythagoras.
The formulas
A cone holds exactly one third of the cylinder with the same base and height. The slant height links the radius and height, and sets the curved (lateral) area.
Worked example
Take a cone with base radius 42 and height 88.
From radius and height, by Pythagoras.
A third of base area times height.
Base circle plus curved side.
What else the calculator returns
You also get the lateral area on its own, the slant height and the base diameter. As with the cylinder, you can enter the diameter on the figure instead of the radius.
Where it comes up
Cone volumes appear in hoppers and funnels, conical piles of sand or aggregate, roof spires and nose cones. The same "one third" rule that makes a cone a third of its cylinder also makes a pyramid a third of its prism.
Frequently asked questions
How do you find the volume of a cone?
Take one third of the base area times the height: V = ⅓πr²h.
Why is a cone one third of a cylinder?
As you move up a cone the circular cross-section shrinks. Adding up those shrinking discs (an integral) comes to exactly a third of the full-size base over the height — the same reason a pyramid is a third of its prism.
What is slant height, and how is it different from height?
Height is the vertical distance from base to apex. Slant height runs up the sloping surface and equals √(r² + h²), so it is always longer than the height.
How do I find the height from the volume?
Rearrange the formula: h = 3V / (πr²).
Does the surface area include the base?
The total surface area does: πr² for the base plus πrl for the curved side. The lateral area on its own is just πrl.