Unroll it, get a sector
Unwrap a cone’s side and you get a circular sector with radius l and arc length 2πr. That’s why L = π r l is “sector area” in disguise.
Net insight
Cone Calculator for Volume, Surface Area, Slant Height, Radius, and Height
Diagram & Inputs
Enter radius and height to calculate every cone measurement.
Lengths
Distance from base center to edge.
Use this instead of radius when known.
Perpendicular height from base to apex.
Edge length from apex to base rim.
Areas
Lateral area plus circular base.
Curved side only, no base.
Volume
Uses cubic units.
Tip: Switch to advanced mode to enter any two values (r/d, h, l, S, L, or V). The calculator checks consistency if you provide more.
Results
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How the Cone Calculator Works
A right circular cone is defined by radius r, height h, and slant height l with l = √(r² + h²). From these:
- Lateral surface area:
L = π r l - Total surface area:
S = L + π r² = π r (l + r) - Volume:
V = (1/3) π r² h
Provide any two values—such as r & h, r & S, h & V, l & S, or L & V. The tool solves for the missing dimensions (using closed-form rearrangements or a quick numeric solve when needed) and derives the rest. If you enter more than two values, it checks them for consistency.
Units: r, d, h, l are lengths; L and S use squared units; V uses cubed units. You can set decimal places and choose a π approximation for classroom alignment.
Practical Cone Examples
Traffic cone with radius 3 and height 4
For r = 3 and h = 4, the slant height is l = √(3² + 4²) = 5. Volume is V = (1/3)π(3²)(4) = 12π ≈ 37.70. Lateral area is 15π ≈ 47.12, and total surface area is 24π ≈ 75.40.
Sheet-metal cone with radius 6 and slant height 10
For r = 6 and l = 10, height is h = √(10² - 6²) = 8. The side material before overlap is the lateral area, L = πrl = 60π ≈ 188.50.
Funnel with volume 50 and height 9
From V = 50 and h = 9, radius is r = √(3V/(πh)) = √(150/(9π)) ≈ 2.30. This is useful when capacity and vertical depth are known first.
Total, Lateral, and Base Area
Total surface area is for a closed cone: curved side plus circular base, S = πrl + πr². Use it for painting or covering the outside when the base is included.
Lateral surface area, also called curved surface area, is the side only: L = πrl. Use it for an open cone, cone wrap, label, or sheet-metal side pattern.
Base area is the circle at the bottom: B = πr². Add it to lateral area only when the cone has a covered base.
Common Mistakes
- Radius vs diameter: if the problem gives diameter, divide by 2 before using formulas with
r. - Height vs slant height: use slant height
lfor lateral area, not vertical heighth. - Missing the one-third factor: cone volume is
(1/3)πr²h, notπr²h. - Mixed units: convert measurements to the same length unit before calculating.
- Total vs lateral area: total area includes the base; lateral area does not.
Formulas & Solvable Pairs (Quick Reference)
- Given r & h:
l = √(r²+h²),L=πrl,S=πr(l+r),V=(1/3)πr²h - Given r & l:
h = √(l²−r²) - Given r & S:
l = S/(πr) − r, thenh = √(l²−r²) - Given r & L:
l = L/(πr), thenh = √(l²−r²) - Given r & V:
h = 3V/(πr²) - Given h & V:
r = √(3V/(πh)) - Given l & S:
r = (−l + √(l² + 4S/π))/2 - Given L & S:
r = √((S−L)/π),l = L/(πr) - Given l & V: solve
(1/3)πr²√(l²−r²) = Vforr(numeric) - Given S & V: solve
πr(r + √(r² + (3V/(πr²))²)) = S(numeric)
Formula Sources and Assumptions
Assumptions: Slant height, lateral area, total surface area, cone angles, and sector-unrolling formulas are for a right circular cone. The volume formula also works for an oblique cone when the perpendicular height is known. Units are labels only; this page does not convert between inches, centimeters, feet, or meters unless you enter converted values.
Sources: These are standard geometry identities. Reference checked against Wolfram MathWorld: Cone. Last reviewed: June 9, 2026.
5 Fun Facts about Cones
Volume is exactly a third
A cone with the same base and height as a cylinder has 1/3 the volume. Archimedes proved it by “filling” a cylinder and sphere together.
Archimedes result
Slant sets everything
Knowing r and slant l fixes area, lateral area, and height in one go. It’s a Pythagorean shortcut often faster than starting with volume.
Quick solve
Angles change circle size
The cone’s apex angle controls how big the unwrapped sector is. A narrow cone makes a tiny “pizza slice”; a wide cone unwraps toward a full circle.
Apex intuition
Spot the frustum
Cut a cone with a plane parallel to its base and you get a frustum. Its volume: V = (1/3)πh(r₁² + r₁r₂ + r₂²)—useful for buckets and cups.
Real-world shape
Cone Calculator: FAQs
How do I calculate cone volume?
Use V = (1/3)πr²h, where r is the base radius and h is the perpendicular height. Enter radius and height, then choose the volume mode or calculate all results.
What is the surface area formula for a cone?
Lateral surface area is L = πrl. Total surface area for a closed cone is S = πrl + πr² = πr(l + r), where l is slant height.
How do I find cone slant height?
For a right circular cone, use l = √(r² + h²). The radius, height, and slant height form a right triangle.
How do I find radius from volume and height?
Rearrange V = (1/3)πr²h to r = √(3V/(πh)). Enter volume and height, then choose find radius.
Why is cone volume one-third of a cylinder?
A cone and cylinder with the same base area and perpendicular height have volumes in a 1:3 ratio, so cone volume is one-third base area times height.
Do these formulas work for oblique cones?
The volume formula works for an oblique cone when the perpendicular height is known. The slant height and surface area formulas here are for right circular cones only.
How do you unroll a cone into a sector?
The lateral surface unwraps into a circular sector with radius l and arc length 2πr. The sector angle in degrees is 360r/l.
Which inputs are valid?
Any two of radius r (or diameter d), height h, slant height l, total surface area S, lateral area L, or volume V. More values are fine; the tool checks consistency.
Does the calculator keep my data private?
Yes. Computation is entirely client-side; nothing is uploaded.
Can I change units, decimal places, or pi?
Yes. Choose a length unit, set decimal places, and pick a π approximation (native precision, 22/7, etc.).