Volume of a Cone Calculator
Calculate the volume, surface area, slant height, and lateral area of a cone from its radius and height. See also Volume of Cylinder Calculator and Volume of Pyramid Calculator.
How to Calculate the Volume of a Cone
To find the volume of a cone, measure the radius of the circular base and the perpendicular height from the base to the apex. Square the radius, multiply by π and the height, then divide by 3. A cone is essentially one-third of a cylinder with the same base and height. This calculator also computes the slant height, total surface area, and lateral surface area.
Cone Volume Formula
V = (1/3) × π × r² × h
Slant Height: l = √(r² + h²)
Lateral Area = π × r × l
Total SA = π × r × (r + l)
Base Area = π × r²
Example
Find the volume of a cone with radius 4 and height 8:
V = (1/3) × π × r² × h
V = (1/3) × π × 4² × 8
V = (1/3) × π × 128
V ≈ 134.0413 cubic units
Slant height l = √(16 + 64) = √80 ≈ 8.9443
Cone Volume Reference Table
| Radius | Height | Volume | Surface Area |
|---|---|---|---|
| 1 | 3 | 3.1416 | 13.0762 |
| 2 | 4 | 16.7552 | 40.6656 |
| 3 | 5 | 47.1239 | 83.2298 |
| 4 | 6 | 100.5310 | 140.8829 |
| 4 | 8 | 134.0413 | 162.6625 |
| 5 | 8 | 209.4395 | 226.7284 |
| 5 | 10 | 261.7994 | 254.1602 |
| 6 | 8 | 301.5929 | 301.5929 |
| 6 | 12 | 452.3893 | 365.9907 |
| 8 | 10 | 670.2064 | 522.9181 |
| 8 | 15 | 1005.3096 | 628.3185 |
| 10 | 10 | 1047.1976 | 758.4476 |
| 10 | 15 | 1570.7963 | 880.5179 |
| 10 | 20 | 2094.3951 | 1016.6407 |
| 15 | 20 | 4712.3890 | 1884.9556 |
Real-World Applications
Ice Cream Cones
Calculate how much ice cream fills a wafer cone — important for manufacturers determining portion sizes and packaging.
Traffic Cones
Engineers calculate the plastic material needed per cone for road safety manufacturing.
Volcanic Mountains
Geologists estimate the volume of volcanic cones to study eruption history and lava output.
Party Hats
Determine the internal space of conical party hats for packaging and material estimation.
Solved Examples
Example 1: Ice Cream Cone Capacity
A wafer cone has a radius of 2.5 cm at the opening and a depth (height) of 12 cm. How much ice cream can it hold?
V = (1/3) × π × 2.5² × 12
V = (1/3) × π × 6.25 × 12
V = (1/3) × π × 75
V ≈ 78.54 cm³ ≈ 78.5 mL
Example 2: Traffic Cone Material
A traffic cone has a base radius of 15 cm and height of 75 cm. What is its volume?
V = (1/3) × π × 15² × 75
V = (1/3) × π × 225 × 75
V ≈ 17,671.46 cm³ ≈ 17.67 liters
Example 3: Sand Pile Volume
A conical pile of sand has a base diameter of 6 m and height of 2 m. Find the volume of sand.
Radius = 6 / 2 = 3 m
V = (1/3) × π × 3² × 2
V = (1/3) × π × 18
V ≈ 18.85 m³
Practice Questions
Q1: Find the volume of a cone with radius 5 cm and height 12 cm.
Answer: V = (1/3) × π × 5² × 12 = (1/3) × π × 300 ≈ 314.16 cm³
Q2: A cone has volume 150 cm³ and height 10 cm. Find its radius.
Answer: r = √(3V / πh) = √(450 / (π×10)) = √(14.32) ≈ 3.78 cm
Q3: What is the slant height of a cone with r=3 and h=4?
Answer: l = √(r² + h²) = √(9 + 16) = √25 = 5 units
Q4: A cylinder and cone have the same base and height. The cylinder volume is 900 cm³. What is the cone volume?
Answer: Cone V = (1/3) × Cylinder V = 900/3 = 300 cm³
Q5: Find the lateral surface area of a cone with r=6 and slant height l=10.
Answer: Lateral SA = πrl = π × 6 × 10 ≈ 188.50 square units
Key Takeaways
- Cone volume = (1/3)πr²h — exactly one-third of a cylinder with the same dimensions.
- The slant height l = √(r² + h²) uses the Pythagorean theorem.
- Unlike cylinders, cones taper to a point, reducing volume by factor of 3.
- Lateral area (πrl) is the curved surface; total SA adds the circular base (πr²).
- The formula works for oblique cones too — just use perpendicular height.
Frequently Asked Questions
What is the volume of a cone?
The volume of a cone is the total three-dimensional space enclosed within its circular base and curved surface tapering to a point (apex). It equals one-third of the volume of a cylinder with the same base and height.
What is the slant height of a cone?
The slant height is the distance from any point on the edge of the circular base to the apex, measured along the surface. It is calculated as l = √(r² + h²) using the Pythagorean theorem.
Why is the cone volume one-third of a cylinder?
This can be proven using calculus (integration) or Cavalieri's principle. Intuitively, three cones with the same base and height can fill exactly one cylinder of the same dimensions.
What is the difference between lateral area and total surface area?
The lateral area is only the curved surface of the cone (πrl). The total surface area adds the circular base (πr²), giving πr(r + l).
Does this formula work for oblique cones?
The volume formula V = (1/3)πr²h works for oblique cones as long as h is the perpendicular height. The surface area and slant height formulas only apply to right circular cones.