Calculate volume, slant height, lateral surface area, total surface area, base area, half-angle, and base circumference of a right circular cone.
A right circular cone is a three-dimensional solid formed by connecting every point on a circle (the base) to a single point (the apex) that lies directly above the center of the base. The word "right" indicates the apex is perpendicular to the base, meaning the axis of the cone is at a right angle to the circular base.
Right circular cones appear everywhere — from ice-cream cones and traffic-safety pylons to volcanic landforms and aerospace nose cones. Understanding the geometry of a cone is essential in civil engineering for earthwork volume calculations, in manufacturing for funnel and nozzle design, and in physics for calculating the moment of inertia of conical objects.
This calculator takes the radius of the base and the perpendicular height and instantly computes seven key measurements: volume, slant height, lateral (curved) surface area, total surface area (including the base), base area, half-angle at the apex, and base circumference. You can quickly switch between common unit systems and explore real-world presets such as a party hat or a traffic cone to see typical values. A surface-area breakdown bar chart visualizes the ratio of lateral area to base area, while the reference table lets you compare common conical objects side by side.
Whether you are a student solving homework problems, an engineer sizing a hopper, or simply curious about the math of cones, this tool gives you accurate, detailed results in seconds.
Cone calculations show up whenever a circular base narrows to a point and both storage capacity and outer material matter. You might need the volume of a hopper, the curved surface area of a paper party hat, or the slant height for a sheet-metal pattern. Those tasks are related, but each one starts from a slightly different measurement, and it is easy to confuse vertical height with slant height if you do the work manually.
This calculator keeps those measurements separate and shows how they interact. That makes it useful for classroom geometry, fabrication estimates, and practical sizing problems where you need more than just the cone volume.
Volume = (1/3)πr²h Slant Height l = √(r² + h²) Lateral SA = πrl Total SA = πrl + πr² Base Area = πr² Half-Angle θ = arctan(r/h) Base Circumference = 2πr
Result: Volume ≈ 314.1593 cm³, Slant Height = 13 cm, Total SA ≈ 282.7433 cm²
With r = 5 cm and h = 12 cm, slant height l = √(25 + 144) = 13 cm. Volume = (1/3)π(25)(12) ≈ 314.16 cm³. Lateral SA = π(5)(13) ≈ 204.20 cm². Base area = π(25) ≈ 78.54 cm². Total SA ≈ 282.74 cm².
The most common cone mistake is treating the vertical height and slant height as if they were interchangeable. They are not. The vertical height runs straight from the base center to the apex, while the slant height lies along the cone's surface. Together with the radius, they form a right triangle, so once you know any two of those measures you can derive the third with the Pythagorean theorem.
In packaging, paper models, and fabricated cones, the lateral surface area often matters more than the volume because it tells you how much material covers the curved side. The total surface area adds the circular base to that curved area. Comparing those two parts is useful when you are coating, wrapping, or cutting a cone from flat stock and need to know where the material is actually going.
The half-angle provides a quick description of how narrow or wide a cone is. Small half-angles create tall, sharp cones, while larger half-angles create broader shapes with more open tops. If you change radius and height while watching the half-angle and slant height together, you can see whether the cone is becoming steeper, more efficient for volume, or more demanding in surface material. That perspective is useful in both geometry problems and real design tradeoffs.
The apex sits directly above the center of a circular base, so the axis is perpendicular (at right angles) to the base. If the apex is offset, the cone is oblique.
Height (h) is the perpendicular distance from the base to the apex. Slant height (l) is the distance along the surface from any point on the base edge to the apex: l = √(r² + h²).
Yes. h = √(l² − r²). This comes directly from the Pythagorean theorem applied to the right triangle formed by r, h, and l.
Cavalieri's principle (or integration) shows a cone's. Understanding this concept helps you apply the calculator correctly and interpret the results with confidence. cross-sectional area tapers linearly to zero; integrating those cross-sections yields exactly 1/3 of the cylinder volume.
The half-angle (θ = arctan r/h) describes how "wide" the cone opens. It appears in optics (numerical aperture), fluid mechanics (Mach cones), and manufacturing (taper angles).
For a frustum with radii R (large) and r (small) and slant height l, the lateral SA = π(R + r)l. This calculator handles full cones; subtract two cones for frustums.