Cone Calculator
Calculate the volume, slant height, lateral surface area, and total surface area of a cone from its base radius and height.
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How to use this calculator
The cone volume is one-third of the enclosing cylinder. Slant height l is the hypotenuse of a right triangle with legs r and h. Lateral surface area unfolds to a sector of a circle.
- 1
Enter the base radius r (distance from center of the base circle to its edge).
- 2
Enter the vertical height h (perpendicular distance from base to apex).
- 3
Volume, slant height, and both surface areas are calculated instantly.
Frequently asked questions
Is “height” the vertical height or the slanted side?
It is the vertical height, measured from the centre of the base straight up to the apex. The slanted length is calculated for you as l = √(r² + h²). If you only know the slant and the radius, get the height first with h = √(l² − r²) and enter that.
What shapes does it cover?
Right circular cones, where the apex sits directly above the centre of a circular base. Oblique cones (leaning), cones with elliptical bases and truncated cones (frustums) need other formulas and are not handled here.
What is the difference between lateral and total surface area?
Lateral area, πrl, is the curved surface only. Total surface area adds the circular base, πr². For r = 3 and h = 4 the lateral area is 47.12389 and the total is 75.398224. Use lateral area for a hollow shape open at the base, such as a funnel cover or party hat, and total area when the base is closed.
What is the “half-angle at apex”?
The angle between the cone’s axis and its slanted side: arctan(r ÷ h). The full opening angle at the tip is twice that. For r = 3 and h = 4 it is 36.869898°, so the full angle is about 73.74°. A radius far bigger than the height gives a flat cone with a half-angle approaching 90°; r = 10, h = 2 gives 78.690068°.
Which units should I use?
Any, as long as radius and height share one. The inputs are labelled “units”. Volume comes out in cubic units and areas in square units. If you enter centimetres, volume is in cm³ and area in cm².
Can I enter the diameter?
Not directly. Halve the diameter first, because the input is the radius. Entering a diameter by mistake makes the volume 4 times too large and the base area 4 times too large.
What happens with zero or negative values?
Radius and height must both be positive. Otherwise the tool returns “Radius and height must be positive” and no results.
How precise are the results?
Calculations use full double precision and the displayed values are rounded to 6 decimal places. The slant height is a hypotenuse, so it is a whole number only for Pythagorean pairs such as 3 and 4 (slant 5) or 5 and 12 (slant 13).
Cone calculator — volume, slant height and surface area
Formulas used
For base radius r and perpendicular height h, the slant height is l = √(r² + h²), the volume is V = ⅓πr²h, the lateral area is πrl, the base area is πr² and the total surface area is their sum. A cone holds exactly one third of the cylinder with the same base and height. The calculator also returns the half-angle at the apex, arctan(r ÷ h), in degrees.
Worked examples
Rows are real output from the calculator.
| r, h | Volume | Slant l | Lateral area | Total area | Half-angle |
|---|---|---|---|---|---|
| 4, 9 | 150.796447 | 9.848858 | 123.764397 | 174.02988 | 23.962489° |
| 3, 4 | 37.699112 | 5 | 47.12389 | 75.398224 | 36.869898° |
| 5, 12 | 314.159265 | 13 | 204.203522 | 282.743339 | 22.619865° |
| 10, 2 | 209.43951 | 10.198039 | 320.380845 | 634.54011 | 78.690068° |
Reading the examples
The 3, 4 cone has a slant height of exactly 5 because 3-4-5 is a right triangle; its volume is 12π. The 5, 12 cone behaves the same way with slant 13 and a volume of 100π, or 314.159265. The last cone is wide and low: its radius is five times its height, so about half of its total area is the base (314.159265 of 634.54011).
Capacity and material questions
For a conical funnel or pile that is 5 units in radius and 12 high, the capacity is 314.159265 cubic units. To estimate the sheet material for a paper cone without a base, use the lateral area (204.203522 for the same cone); add the base area of 78.539816 for a closed one. Remember that real piles of loose material settle at their own slopes, so the right-circular shape is only an approximation.
Input checks and limits
Radius and height must be positive numbers, and the tool assumes the apex is directly above the base’s centre. Use the radius, not the diameter, and keep both inputs in the same unit. If your cone is truncated or leaning, the formulas here do not apply directly. For a cone cut from a sector of a circle, the cone’s slant height is the sector’s radius, and the base circumference equals the sector’s arc length. That relationship lets you recover r and h from a flat pattern before entering them here, which is useful when making paper or sheet-metal cones.
Learn more from an authoritative source:
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Results are estimates for informational purposes only and do not constitute professional financial, medical, legal, or technical advice. Read full disclaimer →