Cone Calculator
Find the volume and surface of a right circular cone from the radius and height, from the radius and slant, or recover r or h from a known volume. The formulas are V = (1/3)πr²h, slant l = √(r² + h²), lateral πrl, and total πr(r + l).
Lateral 47.1238898038 · total S = 75.3982236862
- The base is a circle of area πr² = 3.14159265359 × 3² = 28.2743338823.
- Volume V = (1/3)πr²h = (1/3) × 28.2743338823 × 4 = 37.6991118431.
- Slant l = √(r² + h²) = √(9 + 16) = 5. Lateral = πrl = 47.1238898038. Total S = πr(r + l) = 75.3982236862.
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How to Use the Cone Calculator
From r and h grows a right circular cone. From r and the slant l recovers the height. From V and h recovers the radius. From V and r recovers the height.
- Choose the pair you know.
- Enter non-negative lengths. The slant must be at least the radius. When solving for r or h from volume, the other length must be positive.
- Read the volume (or the missing length), the slant, the lateral area, and the total surface.
A can with the same radius and height belongs on the Cylinder Calculator (V = πr²h, three times this volume). A polygonal base belongs on the Pyramid Calculator (V = (1/3)Bh). A cone with the tip sliced off belongs on the Frustum Calculator (V = (1/3)πh(R² + Rr + r²)). The circular base alone belongs on the Circle Calculator (A = πr²). Converting cubic units belongs on the Volume Converter.
Volume and Surface
A right circular cone is a circular disk drawn up to an apex that sits on the axis. Volume is one-third of the cylinder with the same base and height: V = (1/3)πr²h. That one-third is the same rule as a pyramid (V = (1/3)Bh) when the base is a circle of area πr².
A vertical cross-section through the apex is an isosceles triangle. Half of it is a right triangle with legs r and h, so the slant height — the distance from apex to a point on the rim — is l = √(r² + h²). Unrolling the side gives a sector of radius l whose arc is the circumference 2πr, so the lateral area is πrl. Add the base πr² to get the total S = πr(r + l).
Solving for r from volume: r = √(3V / (πh)). Solving for h: h = 3V / (πr²). From r and l, h = √(l² − r²) provided l ≥ r. Height is perpendicular to the base. An oblique cone still has volume (1/3)Bh with perpendicular height, but its lateral area is not πrl; that case is out of scope. A frustum — this cone with the tip sliced off — belongs on the Frustum Calculator.
Cone Calculator Examples
Radius 3, height 4
l = 5, the 3-4-5 triangle. V = 12π ≈ 37.6991118431. Lateral = 15π ≈ 47.1238898038. Total = 24π ≈ 75.3982236862.
Radius 3, height 5
V = 15π ≈ 47.1238898038. The cylinder of the same r and h, on the Cylinder Calculator, is 45π ≈ 141.372 — exactly three times this cone.
Radius 3, slant 5
h = √(25 − 9) = 4, the first example recovered from the wrapper length instead of the altitude.
Volume of the 3-by-4 cone, known height
Entering V ≈ 37.6991118431 and h = 4 returns r = 3.
Height 0
V = 0 and l = r. The side collapses onto the base. A party hat crushed flat is a disk, not a scoop.
Slant shorter than the radius
l = 2 cannot reach a rim 3 units from the axis. The calculator rejects it instead of inventing a height.
One-Third of the Cylinder
Cavalieri’s picture: a cone and a pyramid of the same base area and height have the same volume, and that volume is one-third of the prism (or cylinder) with the same base and height. For a circle, (1/3) × πr² × h is (1/3)πr²h. The 3-by-5 cone is 15π; the 3-by-5 can is 45π.
Do not drop the 1/3 because “it looks like a can.” Do not use the slant l in place of h in the volume — l is for the wrapper. And do not treat a square-base pyramid as a cone; that solid belongs on the Pyramid Calculator.
Common Cone Mistakes
Using πr²h for a party hat
That is a cylinder, on the Cylinder Calculator. A cone is one-third of that: (1/3)πr²h.
Using the slant as the height in the volume
Volume wants the perpendicular height h. The slant l is for lateral area πrl. On the 3-4-5 cone, plugging l = 5 for h inflates the volume from 12π to 15π.
Forgetting the 1/3
The 1/3 is part of the formula, not optional rounding. A cone is not a cylinder.
Reporting lateral area as the total surface
The wrapper omits the circular lid. Add πr², or use πr(r + l).
Treating a frustum or a pyramid as a cone
A frustum has two circular bases and belongs on the Frustum Calculator. A pyramid has a polygonal base and belongs on the Pyramid Calculator (V = (1/3)Bh).
Converting liters on this page
V is already in the cube of your input unit. Use the Volume Converter for liters and gallons.
Cone Calculator FAQs
What is the volume of a cone?
V = (1/3)πr²h. For r = 3 and h = 4, V = 12π ≈ 37.6991118431 cubic units.
What is the surface area of a cone?
Lateral (the wrapper) is πrl, with l = √(r² + h²). Total surface, including the base, is πr(r + l). For the 3-4-5 cone those are 15π and 24π.
How do you find the radius from the volume?
r = √(3V / (πh)). The height must be positive.
Is a cone one-third of a cylinder?
Yes, when they share the same circular base and the same perpendicular height. The 3-by-5 cone is 15π; the 3-by-5 can is 45π.
Does this treat a frustum, a pyramid, or an oblique cone?
A cut-off tip belongs on the Frustum Calculator. A polygonal base belongs on the Pyramid Calculator. A leaning apex is out of scope. Convert liters on the Volume Converter.
Are my numbers sent to a server?
No. The calculation runs locally in your browser.
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Continue with cylinders, circles, and solid-geometry tools in the SolverCalc collection.
Reviewed by Prof. Dr. Khalil Mudassar (PhD) · Last updated: [mc_modified_date]
Sources
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Disclaimer
This calculator is intended for general educational and practical use. Review important results before relying on them in professional work.