Cone and cylinder volume compared with the sphere
Cone and cylinder volumes with the same base radius r and height h are linked by a simple rule: the cone holds exactly one third of the cylinder, ⅓πr²h against πr²h. Add a sphere of the same radius and make the height h = 2r, and the three volumes stand in the ratio 1 : 2 : 3. With r = 3 in and h = 6 in, the cone holds 56.55 in³, the sphere 113.10 in³ and the cylinder 169.65 in³.
Contents
Cylinder πr²h, cone ⅓πr²h, sphere (4/3)πr³. With r = 3 in and h = 6 in: cylinder 169.65 in³, cone 56.55 in³, sphere 113.10 in³: 1 : 2 : 3.
The comparator above computes the cylinder, the cone with the same base and height, the sphere and the hemisphere of the same radius at once. Set h to twice r to see the 1 : 2 : 3 ratio.
The three volume formulas
| Solid | Volume | r = 3 in, h = 6 in | r = 5 cm, h = 10 cm |
|---|---|---|---|
| Cylinder | πr²h | 169.65 in³ | 785.40 cm³ |
| Cone (same r and h) | ⅓πr²h | 56.55 in³ | 261.80 cm³ |
| Sphere (same r) | (4 ÷ 3)πr³ | 113.10 in³ | 523.60 cm³ |
| Hemisphere (same r) | ⅔πr³ | 56.55 in³ | 261.80 cm³ |
Both columns use h = 2r, so each line is in the ratio 3 : 1 : 2 : 1. In liters, the metric column reads 0.785 L, 0.262 L, 0.524 L and 0.262 L. The method for the cylinder alone is in how to find the volume of a cylinder.
Cone vs cylinder volume: why one third
The cylinder and the cone share the same circular base, πr². The cylinder keeps that full circle all the way up. The cone's slices shrink steadily to a point, and the slice area falls with the square of the distance to the tip. Adding up those shrinking slices gives one third of the full stack, which is the factor ⅓ in V = ⅓πr²h. The same factor applies to every pyramid compared with its prism.
You can check it with water: fill a cone-shaped cup and pour it into a cylindrical cup with the same rim and depth. It takes three pours. In numbers, a cone of r = 3 in and h = 6 in holds 56.55 in³, and 3 × 56.55 = 169.65 in³, the cylinder.
The other way round: to hold as much as a cylinder over the same base, a cone must be three times taller.
Archimedes and the sphere in a cylinder
Put a sphere of radius r snugly inside a cylinder: the cylinder has radius r and height 2r. Its volume is πr² × 2r = 2πr³. The sphere's is (4 ÷ 3)πr³. Divide one by the other: the sphere fills exactly two thirds of its circumscribed cylinder.
Archimedes proved this in his treatise On the Sphere and the Cylinder, together with a matching result for surfaces: the area of the sphere, 4πr², equals the curved side of that cylinder, 2πr × 2r = 4πr². Counting the two end disks as well, the cylinder's total area is 6πr², and the sphere's is again two thirds of it.
Add the cone with the same base and height 2r: ⅓πr² × 2r = ⅔πr³. Now the three volumes are:
| h = 2r | Volume | In units of ⅔πr³ |
|---|---|---|
| Cone | ⅔πr³ | 1 |
| Sphere | (4 ÷ 3)πr³ | 2 |
| Cylinder | 2πr³ | 3 |
Cone + sphere = cylinder: 1 + 2 = 3. The same pattern holds at half the height. With h = r, a cone holds ⅓πr³, a hemisphere ⅔πr³ and a cylinder πr³, again 1 : 2 : 3. For r = 3 in: 28.27, 56.55 and 84.82 in³.
Surface area of a cylinder and cone
Surface areas do not keep the neat volume ratio, because the cone's side depends on its slant height l = √(r² + h²).
| Solid | Total surface area | r = 3 in, h = 6 in | r = 5 cm, h = 10 cm |
|---|---|---|---|
| Cylinder | 2πr² + 2πrh | 169.65 in² | 471.24 cm² |
| Cone | πr² + πrl | 91.50 in² (l = 6.708 in) | 254.16 cm² (l = 11.180 cm) |
| Sphere | 4πr² | 113.10 in² | 314.16 cm² |
With h = 2r, the sphere's area is two thirds of the cylinder's, as Archimedes found. The cone has no simple ratio. More on the cylinder side is on the page about the surface area of a cylinder.
Word problems on cylinders, cones and spheres
- Three cones into a cylinderA cylindrical glass has r = 4 cm and h = 12 cm. How many full cone-shaped cups with r = 4 cm and h = 12 cm does it take to fill it? Answer: 3, whatever the size, since each cone is ⅓ of the cylinder.
- Melting a sphereA metal ball of radius 3 in is melted and recast as a cylinder of radius 3 in. How tall is it? (4 ÷ 3)π × 27 = π × 9 × h, so h = 4 in.
- Balls in a canThree balls of radius 1.3 in fit snugly in a can of radius 1.3 in and height 7.8 in. The can holds π × 1.69 × 7.8 ≈ 41.41 in³, the balls 3 × (4 ÷ 3)π × 1.3³ ≈ 27.61 in³: two thirds. The empty space is about 13.80 in³.
- Cone with a hemisphere on topA cone of r = 1.5 in and h = 5 in topped with a hemisphere of the same radius holds ⅓π × 2.25 × 5 + ⅔π × 3.375 ≈ 11.78 + 7.07 = 18.85 in³.
A cylinder topped with a dome or a cone is worked out step by step on the page about a cylinder with a hemisphere on top. For more exercises with answers on the cylinder alone, see the cylinder volume practice problems.
Common mistakes
- Applying ⅓ when the heights differ. The one-third rule needs the same base and the same height.
- Using h in the sphere formula. A sphere has only a radius: (4 ÷ 3)πr³.
- Using the slant height for the cone's volume. The volume uses the perpendicular height h; the slant l is only for the surface area.
- Expecting 1 : 2 : 3 for any height. The sphere matches only when h = 2r (or a hemisphere when h = r).
Remember the stack: cone + sphere = cylinder, when all three share the radius and the cylinder is as tall as the sphere.
For the cylinder on its own, in any unit, use the cylinder volume calculator.
FAQ
What is the relationship between the volume of a cone and a cylinder?
A cone holds exactly one third of a cylinder with the same base radius and the same height: V(cone) = ⅓πr²h, V(cylinder) = πr²h. Three full cones fill the cylinder.
Why is the ratio of cone, sphere and cylinder 1:2:3?
Take a sphere of radius r and the cylinder that just contains it (height 2r), and a cone with the same base and height. Then the cylinder is 2πr³, the sphere (4 ÷ 3)πr³ and the cone ⅔πr³. Dividing by ⅔πr³ gives 1 : 2 : 3.
How do you find the volume of cylinders, cones and spheres?
Cylinder: V = πr²h. Cone: V = ⅓πr²h. Sphere: V = (4 ÷ 3)πr³. Use the radius, not the diameter, and the perpendicular height. Results are in cubic units.
What is the surface area of a cylinder and a cone?
Cylinder: 2πr² + 2πrh (two disks and the side). Cone: πr² + πrl, where l = √(r² + h²) is the slant height. With r = 3 in and h = 6 in, the cylinder has about 169.65 in² and the cone about 91.50 in².
Which holds more, a cone or a cylinder of the same size?
The cylinder, three times as much, when the base and the height are the same. A cone needs three times the height of a cylinder to hold the same volume over the same base.
What fraction of a cylinder does a sphere fill?
Exactly two thirds when the cylinder just contains the sphere (same radius, height equal to the diameter). Three balls stacked in a tight can fill two thirds of it, because the can is three such cylinders stacked.
Related
- How to find the volumeHow to find the volume of a cylinder: square the radius, multiply by π and the height. Worked examples in inches, feet, cm and m, plus common mistakes.
- Cylinder + hemisphereVolume of a cylinder with a hemisphere on top: V = πr²h + ⅔πr³. Capsules with hemispherical ends, a cone on top, partly filled tanks, worked examples.
- Surface area of a cylinderThe surface area of a cylinder is A = 2πr² + 2πrh. Get the total, lateral and base areas from a radius or diameter, with worked examples in inches and cm.
Updated October 12, 2026