Cone Calculator
Volume, surface area and slant height of a cone.
Volume
75.3982
r 3, h 8
Surface area
108.7997
Slant height
8.544
AI Breakdown & Smart Takeaway
Plain-English insight on your numbers
Get a personalized explanation of what these results mean — and how to improve them.
How the Cone Calculator works
The GKCalculators Cone Calculator instantly computes the volume, total surface area, lateral surface area, and slant height of any right circular cone from just two inputs — radius and height. It's built for students, engineers, architects, and anyone working with conical shapes in math, design, or manufacturing.
A right circular cone is a three-dimensional geometric solid with a circular base and a single apex point directly above the center of that base. The two primary measurements you need are the base radius (r) and the perpendicular height (h) — the straight vertical distance from the base to the apex. From these two values, every other property of the cone can be derived through well-established geometry formulas, which is exactly what this calculator automates.
The slant height (l) is the diagonal distance from the edge of the base circle to the apex, running along the outer surface of the cone. It is not the same as the vertical height, and confusing the two is one of the most common mistakes in cone geometry. The slant height is calculated using the Pythagorean theorem: l = √(r² + h²). It matters enormously for surface area calculations because the lateral (side) surface is measured along the slant, not the vertical.
Volume measures the three-dimensional space enclosed inside the cone, calculated as V = (1/3)πr²h. Notice the one-third factor — a cone holds exactly one-third the volume of a cylinder with the same base and height. Surface area has two components: the lateral surface area (the curved outer face), given by πrl, and the base area, which is simply πr². The total surface area adds these together: SA = πr² + πrl = πr(r + l). This calculator computes all three — lateral area, base area, and total surface area — separately so you can use whichever value your problem requires.
A frequent source of error is mixing up diameter and radius. The radius is half the diameter, so if you measure across the full width of a circular base, be sure to halve that value before entering it. Units also matter: if your radius is in centimeters and your height is in centimeters, volume will come out in cubic centimeters and area in square centimeters. This calculator keeps units consistent, but always verify your inputs are in the same unit system before calculating, especially in real-world applications like construction or product design.
Formula
V = 1/3·πr²h · slant = √(r² + h²)
Pro tips
- Always confirm whether you have the radius or the diameter of the base — if you measured across the full circle, divide by 2 before entering the value, as entering the diameter instead of the radius will quadruple your volume result.
- If you know the slant height and radius but not the vertical height, rearrange the Pythagorean relationship: h = √(l² − r²). Many real-world cone measurements (like a party hat or funnel) are easier to measure along the slant than the true vertical.
- For material estimation projects (painting a conical roof, cutting sheet metal), use only the lateral surface area, not the total surface area — the base is often already accounted for or is open.
- When scaling a cone design, remember that volume scales with the cube of the scale factor while surface area scales with the square. Doubling both radius and height gives 8× the volume but only 4× the surface area.
- For oblique cones (where the apex is not directly above the center), these formulas don't apply directly. This calculator is specifically designed for right circular cones, which are the standard form used in most academic and engineering contexts.
Key terms
- Radius (r)
- — The distance from the center of the cone's circular base to its edge; equal to half the diameter of the base.
- Height (h)
- — The perpendicular (vertical) distance from the center of the base to the apex of the cone, measured at a right angle to the base plane.
- Slant Height (l)
- — The straight-line distance measured along the outer surface of the cone from the base edge to the apex, calculated as l = √(r² + h²).
- Lateral Surface Area
- — The area of the curved outer surface of the cone, excluding the circular base, given by πrl.
- Total Surface Area
- — The complete outer area of the cone including both the curved lateral surface and the circular base, given by πr(r + l).
- Volume
- — The three-dimensional space enclosed within the cone, equal to one-third the product of the base area and height: V = (1/3)πr²h.
