How to use the cone volume calculator
- Enter the radius of the cone's circular base.
- Enter the height, which is the perpendicular distance from the base to the tip.
- The results update instantly, showing the volume, slant height and surface areas.
- Change either value to see how the geometry responds.
How the cone volume is calculated
The volume of a right circular cone equals one third of the base area multiplied by the height. The formula is:
V = (1/3) × π × r² × h
For a cone with a radius of 3 and a height of 4, the calculation proceeds as follows. First square the radius: 3² = 9. Multiply by pi: 9 × 3.14159... = 28.2743. Multiply by the height: 28.2743 × 4 = 113.0973. Finally divide by three: 113.0973 / 3 = 37.6991. This cone volume calculator performs that arithmetic for any positive inputs you provide.
Slant height and surface area formulas
A cone has two distinct surface components: the flat circular base and the curved lateral side. To find the lateral area you first need the slant height, which is the straight-line distance from the edge of the base to the apex. By the Pythagorean theorem:
l = √(r² + h²)
With r = 3 and h = 4, the slant height is √(9 + 16) = √25 = 5 exactly. The lateral surface area then uses the formula L = π × r × l, giving 3.14159... × 3 × 5 = 47.1239. The base area is simply B = π × r² = 28.2743. Adding them yields the total surface area of 75.3982.
Common cone dimensions reference
The table below shows pre-computed values for several standard cones. All figures are rounded to four decimal places.
| Radius | Height | Slant height | Volume | Total surface area |
|---|---|---|---|---|
| 1 | 1 | 1.4142 | 1.0472 | 7.5845 |
| 2 | 5 | 5.3852 | 20.9440 | 46.4537 |
| 3 | 4 | 5.0000 | 37.6991 | 75.3982 |
| 5 | 12 | 13.0000 | 314.1593 | 282.7433 |
| 10 | 10 | 14.1421 | 1047.1976 | 758.4475 |
Practical uses for cone calculations
Knowing the volume of a cone is useful in many trades and sciences. Traffic cones, funnels, conical flasks in chemistry labs and ice cream cones all share this shape. In construction, conical piles of gravel or sand can be estimated using the same formula, though for bulk material planning you may also want a dedicated gravel calculator that accounts for compaction and density.
In manufacturing, sheet metal workers use the lateral surface area to determine how much material is needed to roll a conical duct or hopper. The slant height tells them the radius of the flat pattern before rolling. For very large or very small measurements, pairing this tool with a scientific notation calculator helps keep numbers readable.
Edge cases and common mistakes
A frequent error is confusing the diameter with the radius. If you measure across the base, divide by two before entering the value. Another mistake is using the slant height instead of the perpendicular height in the volume formula. The height must be measured at a right angle to the base, not along the sloped side.
When the height is zero, the cone collapses into a flat disk. The volume becomes zero, but the total surface area equals the base area because the lateral side vanishes. When the radius is zero, the shape degenerates into a line segment with no volume and no surface area. This cone volume calculator handles both limits correctly.
Worked examples
Each result below was produced by the same code the tool runs, so you can check your own numbers against them.
| Example | Input | Result |
|---|---|---|
| Classic 3-4-5 cone | Radius (r): 3; Height (h): 4 | Volume: 37.6991118431; Slant height: 5; Lateral surface area: 47.1238898038; Base area: 28.2743338823; Total surface area: 75.3982236862 |
| Tall narrow cone | Radius (r): 2; Height (h): 10 | Volume: 41.8879020479; Slant height: 10.1980390272; Lateral surface area: 64.0761689777; Base area: 12.5663706144; Total surface area: 76.642539592 |
| Wide shallow cone | Radius (r): 8; Height (h): 3 | Volume: 201.06192983; Slant height: 8.54400374532; Lateral surface area: 214.734235188; Base area: 201.06192983; Total surface area: 415.796165018 |
| Unit cone | Radius (r): 1; Height (h): 1 | Volume: 1.0471975512; Slant height: 1.41421356237; Lateral surface area: 4.44288293816; Base area: 3.14159265359; Total surface area: 7.58447559175 |
FAQ
What is the formula for the volume of a cone?
The volume equals one third of pi times the radius squared times the height: V = (1/3) * pi * r^2 * h.
How do I find the slant height of a cone?
Use the Pythagorean theorem: l = sqrt(r^2 + h^2), where r is the base radius and h is the perpendicular height.
Is the volume of a cone exactly one third of a cylinder?
Yes. A cone with the same base radius and height as a cylinder holds exactly one third of the cylinder's volume.
Can I use this cone volume calculator if I only know the diameter?
Divide the diameter by two to get the radius, then enter that value along with the height.
Does this tool work for oblique cones?
The volume formula V = (1/3) * pi * r^2 * h applies to oblique cones as well, provided h is the perpendicular height. However, the surface area formulas shown here assume a right circular cone.
What units does the calculator use?
The calculator is unitless. If you enter the radius and height in centimeters, the volume will be in cubic centimeters and the areas in square centimeters.