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🧊 Cylinder & Cone Volume Lab

Same Base, Same Height —
Yet the Volume Differs by Exactly 3×

Change the radius and height with the sliders. Place a cylinder and a cone side by side with the same base radius and height, and the cone's volume is always exactly 1/3 of the cylinder's.

Imagine filling an ice cream cone all the way up, then pouring it into a cylindrical cup with the same radius and height. How many pours would 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.

What's remarkable is that this 3-to-1 relationship never breaks, no matter how much the radius or height changes. Whether the shapes get bigger or smaller, as long as their base and height match, the ratio always stays exactly the same.

Change the radius and height with the sliders and check for yourself that the two volume numbers always differ by exactly a factor of 3.

Cylinder Cone
Radius (r)4cm
Height (h)8cm
Cylinder volume
402.1
Cone volume
134.0
Cone ÷ Cylinder = 0.333... (always 1/3!)

🧩 Why Exactly 1/3? — See It Yourself With a Triangular Prism

You can see with your own eyes, without any calculation, why a pyramid's volume is always 1/3 of a prism's. A single triangular prism can be split into 3 tetrahedra (triangular pyramids) of exactly equal volume — press the buttons and split it apart yourself.

🔄 Drag the diagram to rotate it to any angle you like · Reset angle
Pyramid 1 Pyramid 2 Pyramid 3
Split a triangular prism into three pyramids this way, and all three pyramids' volumes are exactly equal (even though their shapes differ slightly, it can be proven by calculation that their volumes match). So one pyramid's volume is 1/3 of the prism's, and since the prism's volume is base area × height, the pyramid's volume naturally works out to "(1/3) × base area × height."
This same principle carries straight over to cones and cylinders. Instead of a pyramid with a triangular base, keep increasing the number of sides — 4, 5, and on and on — and the base gets closer and closer to a circle, while the pyramid gets closer and closer to a cone. No matter the shape of the base, the relationship "pyramid volume = (1/3) × base area × height" always holds, so even once the base becomes a perfect circle — a cone — that exact 1/3 ratio stays exactly the same.