Cylinder vs Cone Volume Calculator
A cylinder has three times the volume of a cone with matching dimensions using V cone = 1/3 pi r squared h. For a radius of 6 cm and height 10 cm, the cylinder holds 1,130.97 cm3 while the cone holds 376.99 cm3. Compare volume ratios and capacities side by side below.
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Volume: Cylinder vs Cone
A cone with the same radius and height as a cylinder has exactly one-third (⅓) the volume. This relationship is a fundamental property of 3D solids.
The formulas show the relationship:
- Cylinder volume = π × r² × h
- Cone volume = (1/3) × π × r² × h
- Ratio: Cone / Cylinder = 1/3
3 cones with identical radius and height fill exactly 1 cylinder. This can be demonstrated by filling a cone with water three times and pouring the water into a cylinder of the same dimensions: the cylinder fills completely.
Cylinder vs Sphere Volume
A sphere inscribed inside a cylinder (touching both bases and the side) has a specific volume relationship to that cylinder. The sphere volume equals two-thirds (⅔) of the cylinder volume.
The formulas are:
- Cylinder volume = π × r² × h = π × r² × 2r = 2πr³
- Sphere volume = (4/3) × π × r³ = (4/3)πr³
- Ratio: Sphere / Cylinder = (4/3)πr³ / 2πr³ = 2/3
For a cylinder with radius 5 cm and height 10 cm (2r): cylinder volume = 2π × 125 = 785.40 cm³. The inscribed sphere volume = (4/3) × π × 125 = 523.60 cm³, which is exactly ⅔ of 785.40.
Geometric volume comparisons and 3 to 1 ratio
This tool compares a cylinder to a cone with the same base and height. A cone holds exactly one third the volume of the cylinder.
The cone, cylinder, and sphere relationship
Cylinder volume is V = pi r squared h. Cone volume is V = (1/3) pi r squared h. Three matching cones fill one cylinder completely.
Frequently Asked Questions
What is the exact ratio of a cone to a cylinder?
If both a cone and a cylinder have the exact same base radius and the exact same height, the cylinder will always hold exactly 3 times more volume than the cone. The ratio is 3:1.
How do you find the volume of a cone?
The formula for a right circular cone is V = ⅓ × π × r² × h. You simply calculate what would be the volume of a tall cylinder, then divide it by three.
Why is a cone 1/3 the volume of a cylinder?
This is a fundamental theorem of calculus and geometry. When integrating the cross-sectional area of a cone along its height, the volume mathematically reduces to exactly one-third of its bounding cylinder.
What if I include a sphere in the comparison?
Archimedes famously proved the relationship between all three. If a sphere, cone, and cylinder all share the same radius (and the height is 2r), the volumes are in a perfect 1:2:3 ratio. The cone is 1, the sphere is 2, and the cylinder is 3.
Are the surface areas also a 1:3 ratio?
No! The 1:3 ratio only applies to internal volume. Surface area depends heavily on the slant height of the cone, which requires calculating Pythagorean values, breaking the clean 1:3 ratio.
Why is the volume of a cone one-third of a cylinder?
The volume of a cone is one-third of a cylinder with the same radius and height because of how the cone's cross-sectional area decreases from base to apex. At height y from the base, the cone's cross-sectional radius is r × (h − y)/h, giving an area of π × r² × ((h − y)/h)². Integrating this area from 0 to h yields (1/3) × π × r² × h: exactly one-third of the cylinder volume πr²h. This can be physically demonstrated by filling a cone with water 3 times and pouring the water into a cylinder of the same dimensions.