Question

If the height and the radius of both containers are the same, do you think the contents of three cones will always fill the cylinder exactly?
Cylinder
Radius
Height
2 in.
4 in.
2 in.
4 in.
4 in.
8 in.
Cone
Radius
2 in.
4 in.
2 in.
Height
4 in.
3
B
4 in.
?
8 in.
?
AZ
How many cones fill the cylinder?
Yes
A
No
C

Answer 1

Yes — three identical right circular cones will exactly fill one cylinder when the cone and the cylinder have the same radius r and the same height h.

Reason: Volume of the cylinder = πr^2h. Volume of one cone = (1/3)πr^2h. So three cones have volume 3·(1/3)πr^2h = πr^2h, which equals the cylinder's volume.

Example: r = 2 in, h = 4 in:
- Cylinder: π(2^2)(4) = 16π in^3
- Cone: (1/3)·16π = 16π/3 in^3
- Three cones: 3·16π/3 = 16π in^3, which exactly fills the cylinder.

(If the radii or heights differ, the equality no longer holds.)