Fill an ice cream cone all the way up, then pour it into a cylindrical cup with exactly the same radius and height — how many pours does it take to fill the cup? The answer is always exactly 3. A cone's volume is exactly 1/3 of a cylinder's volume with the same base and height. It's a remarkable property that this relationship never changes, no matter how much the cone's and cylinder's overall size differ, as long as the base radius and height match.
By formula, a cylinder's volume is "base area × height," or V = πr²h. That's the circle's area (πr²) multiplied by the height, which makes natural sense once you picture a cylinder as a stack of very thin circular disks. A cone's volume is that formula times 1/3: V = (1/3)πr²h. It's also known that pyramids with the same base and height (square pyramids, triangular pyramids, and so on) are similarly exactly 1/3 of the corresponding prism's volume — this is a general property that applies to cone-and-pyramid-shaped solids as a whole.
Why exactly 1/3? A rigorous proof requires calculus to fully understand, but there's a way to build intuition for it. It's known that a triangular prism with a given base and height can be split into exactly 3 triangular pyramids, and showing that these 3 pyramids all have equal volume naturally gives you the relationship "pyramid volume = prism volume ÷ 3." Think of a cone and cylinder as the curved version of this triangular-pyramid-to-triangular-prism relationship, and it starts to click why the ratio is 1/3.
This 1/3 rule comes in handy in daily life too. Piles of dirt or sand at a construction site naturally settle into a cone shape, and you can calculate their volume right away just by measuring the base radius and height. Funnels, paper cups, and party hats are all cone-shaped, so this formula applies to them too. The same principle is used, in reverse, when calculating the capacity of cylinder-shaped facilities like grain silos or water tanks. Knowing that calculating a pyramid's volume uses the exact same formula, base area × height ÷ 3, also makes it easy to estimate even the Egyptian pyramids' enormous volume.
When studying this with kids, the most effective activity is actually filling a cylinder and cone model (or a model you make out of paper yourselves) that have matching volumes, with water, sand, or rice, and pouring it back and forth. Fill the cone and pour it into the cylinder, and confirm for yourself that it only fills 1/3, taking three pours to fill it up — the formula feels a lot more trustworthy once you've seen that firsthand. Building several cylinder-cone pairs with different radii or heights and checking that this 3-to-1 relationship always holds is also a great follow-up activity.
On our activity page, adjust the radius and height with sliders and watch the volumes of a matching-size cylinder and cone calculate in real time, always differing by exactly a factor of 3. Trying different values and checking for yourself "is it really 3 times this time too?" is also a fun way to verify it.