Cylinder and Cone Calculator: Volume, Surface Area, Slant Height and Radius from Volume

The cylinder and cone calculator puts two solids with the same circular base into one panel. Enter base radius r and height h to get cylinder volume \pi r^2h, cylinder total surface area 2\pi r(r+h), cone volume \tfrac13\pi r^2h, and cone total surface area \pi r(r+l), where the slant height l=\sqrt{r^2+h^2} is derived automatically. If cylinder volume and height are known, the calculator can solve for radius.

Cylinder and Cone Calculator: Volume, Surface Area, Slant Height and Radius from Volume

About this calculator

The cylinder and cone calculator puts two solids with the same circular base into one panel. Enter base radius $r$ and height $h$ to get cylinder volume $\pi r^2h$ , cylinder total surface area $2\pi r(r+h)$ , cone volume $\tfrac13\pi r^2h$ , and cone total surface area $\pi r(r+l)$ , where the slant height $l=\sqrt{r^2+h^2}$ is derived automatically. If cylinder volume and height are known, the calculator can solve for radius.

Typical cylinder-like objects include cans, pipes, tanks and silos. Funnels, traffic cones and conical roofs are modeled as cones. Because both shapes share the same circular base, their formulas are easy to compare and combine.

This calculator assumes right circular cylinders and right circular cones. It does not directly handle frustums, oblique solids, hollow-wall volume or unit conversion.

Formula and variables

V_{\text{cyl}} = \pi r^2 h,\qquad S_{\text{cyl}} = 2\pi r^2 + 2\pi r h = 2\pi r(r+h)

V_{\text{cone}} = \tfrac13\pi r^2 h,\qquad l = \sqrt{r^2+h^2},\qquad S_{\text{cone}} = \pi r^2 + \pi r l = \pi r(r+l)

Solving radius from cylinder volume:

r = \sqrt{\frac{V_{\text{cyl}}}{\pi h}}

Symbol	Meaning	Unit	Notes
$r$	base radius	length	positive; radius, not diameter
$h$	perpendicular height	length	positive
$l$	cone slant height	length	$\sqrt{r^2+h^2}$
$V_{\text{cyl}}$	cylinder volume	length³	base area × height
$S_{\text{cyl}}$	cylinder total surface area	length²	includes both bases
$V_{\text{cone}}$	cone volume	length³	one third of same-base, same-height cylinder
$S_{\text{cone}}$	cone total surface area	length²	includes one circular base

The cylinder lateral area is $2\pi rh$ . The cone lateral area is $\pi rl$ .

How to use this calculator

Open “Math tools → Formulas → Cylinder / cone” (/#/mathtools/formula?calculator=calculator.cylinder-cone).
Enter radius and height; all four outputs update together.
If you measured a diameter, divide it by 2 before entering it as $r$ .
To solve for radius, clear the radius field, enter cylinder volume and keep height. The radius field becomes computed from $\sqrt{V/(\pi h)}$ .
In radius-solve mode the other outputs may temporarily show 0 because the engine evaluates them with the missing radius treated as zero. Enter the solved radius back into the radius field to obtain the complete set of outputs.
If you need lateral area only, subtract the base area from total surface area: two bases for a cylinder, one for a cone.

Worked examples

Example 1: r = 3, h = 10

$\pi r^2 = 9\pi = 28.27433388$ .

Cylinder volume: $90\pi = 282.74333882$ .
Cylinder surface area: $78\pi = 245.04422698$ .
Cone volume: $30\pi = 94.24777961$ .
Cone slant height: $\sqrt{109}=10.44030651$ .
Cone surface area: $\pi\cdot3(3+10.44030651)=126.67190457$ .

Example 2: the 3-4-5 cone

For $r=3$ , $h=4$ , the slant height is exactly $l=5$ .

Cylinder volume: $36\pi = 113.09733553$ .
Cylinder surface area: $42\pi = 131.94689145$ .
Cone volume: $12\pi = 37.69911184$ .
Cone surface area: $24\pi = 75.39822369$ .

The cone lateral area is $15\pi$ , and its unfolded lateral surface is a sector of radius 5 whose arc length is the base circumference $6\pi$ .

Figure 1: cylinder dimensions, a 3-4-5 cone, and the cone lateral surface unfolded as a circular sector

Example 3: solve radius from V = 1000 and h = 10

r = \sqrt{\frac{1000}{10\pi}} = 5.64189584

If the length unit is centimeters, this corresponds to a 1 L cylinder about 11.28 cm in diameter and 10 cm tall.

Use the 3-4-5 example to compare the cylinder and cone with the same base and height. The cone volume should remain exactly one third of the cylinder volume.

Principle and derivation

A cylinder is a stack of identical circular cross-sections, so its volume is base area times height: $\pi r^2h$ . Its surface consists of two circles plus a rectangle formed by unrolling the side, with width $2\pi r$ and height $h$ .

A right cone has the same circular base but tapers linearly to a point. Its volume is one third of the corresponding cylinder. The lateral surface unrolls into a circular sector whose radius is the slant height $l$ and whose arc length equals the base circumference $2\pi r$ . Sector area is therefore

\frac12\,(2\pi r)\,l = \pi rl.

The slant height comes from the right triangle through the cone axis: $l^2=r^2+h^2$ .

Assumptions

The solid is right and circular.
Radius and height use the same length unit.
Surface area means total external area unless you explicitly subtract one or more bases.
Inputs are geometric dimensions, not wall thicknesses.

Limitations

Supported:: cylinder and cone volume and total surface area for positive $r$ and $h$ , plus cylinder radius from volume and height.
Approximate only:: real containers may have wall thickness, rounded corners, deformation, and nominal capacities that differ from ideal geometric volume.
Not covered:: frustums, oblique cylinders, hollow-wall volume, inverse cone calculations, or inverse calculations from surface area.
Intermediate geometry:: slant height, lateral area, base area, and net-sector angle are not separate calculator outputs.
Numerics:: very large or very small dimensions may display in scientific notation.

Common pitfalls

Entering diameter as radius doubles $r$ and multiplies volume by 4.
Using height $h$ instead of slant height $l$ in cone lateral area.
Forgetting one or both bases when computing total surface area.
Mixing centimeters and meters before squaring or cubing.
Expecting a frustum formula from a full-cone calculator.

Typical use cases

Container capacity

Estimate the volume of cups, tanks, drums, silos, or combined cylinder-and-cone shapes; solve a cylinder radius for a target volume.

Material estimates

Estimate sheet area for cylindrical tanks, coating area, or material for conical covers and funnels.

Engineering and earthwork

Approximate conical stockpile volume or cylindrical concrete volume from measured dimensions.

Teaching

Explore the one-third cone/cylinder volume relationship, square/cube scaling, and the connection between cone nets and circular sectors.

FAQ

Why is a cone one third of the matching cylinder?

For the same base and height, Cavalieri-style cross-section arguments and classical geometry show the cone occupies exactly one third of the cylinder volume.

How do I get an open-top cylinder area?

Compute total cylinder surface area and subtract one base area $\pi r^2$ .

How do I convert cubic centimeters to liters?

$1\,\text{cm}^3 = 1\,\text{mL}$ and $1000\,\text{cm}^3 = 1\,\text{L}$ .

Privacy

All inputs and calculations stay in your browser. This static explanation does not record your values.

Sources & review

Reviewed by: CalcX 编辑组
Updated: 2026-09-09

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