Discover why a cylinder holds exactly 3 times the volume of a cone with the same dimensions. Complete comparison with formulas, mathematical proof, and real-world applications.
Vcylinder = 3 × Vcone
One of the most important relationships in geometry: a cylinder holds exactly 3 times the volume of a cone with the same base radius and height. This 3:1 ratio is fundamental in engineering, construction, and manufacturing calculations.
Key Formula Relationship:
Vcylinder = 3 × Vcone
When radius and height are equal, the cylinder always contains three cones worth of volume.
V = πr²h
where r = radius, h = height
V = ⅓πr²h
where r = radius, h = height
The cone tapers from a full circular base to a single point at the top. As you move up the cone, each circular cross-section gets smaller. The average of all these circles equals exactly 1/3 of the base area, which is why the volume formula includes the ⅓ factor.
Mathematical proof: Using calculus integration from 0 to h, the cone's volume integrates to ⅓πr²h, proving this elegant relationship.
| Feature | Cylinder | Cone |
|---|---|---|
| Shape | Straight sides, uniform diameter | Tapers from base to point |
| Volume Formula | V = πr²h | V = ⅓πr²h |
| Volume Ratio | 3× (reference) | 1× (1/3 of cylinder) |
| Surface Area | 2πr² + 2πrh | πr² + πr√(r²+h²) |
| Cross-sections | Constant circles (same size) | Circles decreasing to point |
| Stability | More stable (uniform weight) | Less stable (top-heavy possible) |
| Storage Efficiency | High (no wasted space) | Lower (tapered design) |
| Material Use | More material needed | Less material (saves cost) |
| Flow/Drainage | Uniform (no natural flow) | Excellent (gravity-assisted) |
A cylinder and cone both have radius = 4 feet and height = 6 feet. How do their volumes compare?
V = πr²h
V = π × 4² × 6
V = π × 16 × 6
V = 96π ≈ 301.59 ft³
V = ⅓πr²h
V = ⅓ × π × 4² × 6
V = ⅓ × π × 16 × 6
V = 32π ≈ 100.53 ft³
Verification of 3:1 Ratio:
301.59 ÷ 100.53 = 3.00 (exact 3:1 relationship confirmed!)
A cylindrical water tank: radius 3 m, height 5 m
Given: r = 3 m, h = 5 m
V = π × 3² × 5
V = π × 9 × 5
V ≈ 141.37 m³
Capacity: ~141,370 liters
A traffic cone: base radius 0.15 m, height 0.7 m
Given: r = 0.15 m, h = 0.7 m
V = ⅓ × π × 0.15² × 0.7
V = ⅓ × π × 0.0225 × 0.7
V ≈ 0.0165 m³
Capacity: ~16.5 liters
Cylinders:
Cones:
Cylinders:
Cones:
Cylinders:
Cones:
Cylinders:
Cones:
Using V = πr²h for a cone gives you the cylinder volume (3× too much).
✓ Solution: Always include the ⅓ (or divide by 3) for cone volume calculations.
Both formulas require radius (r), not diameter (d). Using diameter makes volume 4× too large.
✓ Solution: If given diameter, divide by 2 first: r = d ÷ 2, then calculate volume.
For cones, use vertical height (h), not slant height (l). Slant height is for surface area, not volume.
✓ Solution: If given slant height l: h = √(l² - r²) using Pythagorean theorem.
Volume units are cubic (ft³, m³), but inputs may be in different units (inches, cm, gallons).
✓ Solution: Convert all dimensions to same unit before calculating. Use our unit converters!
A water park is designing a water slide landing pool. They have two options with same material cost:
Which pool actually holds more water?
Cylinder calculation:
V = πr²h = π × 4² × 2 = 32π ≈ 100.53 m³
Cone calculation:
V = ⅓πr²h = ⅓ × π × 4² × 6 = 32π ≈ 100.53 m³
Answer: Both hold exactly the same volume! The cone needs to be 3× deeper (6 m vs 2 m) to compensate for its tapered shape. This demonstrates the perfect 3:1 relationship.
As a cone tapers from base to point, each horizontal cross-section is a progressively smaller circle. The average radius across all heights equals r/√3, which when integrated over the full height mathematically produces the ⅓ factor. This can be proven using calculus or demonstrated by physically filling three identical cones to fill one cylinder.
Yes! To find a cone with equal volume to a cylinder (same radius), the cone must be 3× taller. Example: A cylinder r=5, h=4 has volume ≈ 314 units³. An equivalent cone with r=5 needs h=12 to achieve the same volume: V = ⅓π(5²)(12) ≈ 314 units³.
Cylinders are more storage-efficient, holding 3× more volume in the same base footprint and height. However, cones are more material-efficient (use 67% less material) and better for gravity-fed systems. Choose based on your priority: maximum capacity (cylinder) or material savings/flow (cone).
First convert diameter to radius (r = d ÷ 2), then use the formula. Example: diameter = 10 ft, height = 8 ft. Radius = 5 ft. Cylinder: V = π(5²)(8) ≈ 628 ft³. Cone: V = ⅓π(5²)(8) ≈ 209 ft³.
Vertical height (h) is the perpendicular distance from base to apex - use this for volume calculations. Slant height (l) is the distance along the cone's surface from base edge to apex - use this for surface area. They're related by: l = √(r² + h²). Always use vertical height for volume formulas.
Grain silos combine cylindrical tops (maximum storage) with conical bottoms (easy discharge). The cone shape uses gravity to funnel all grain to a central outlet, preventing material from getting stuck. This hybrid design maximizes both storage capacity and operational efficiency.