How do you find the volume and surface area of a cylinder?

The volume of a cylinder is V = πr²h (base area times height). The lateral (curved) surface area is L = 2πrh. The total surface area is T = 2πrh + 2πr² = 2πr(h + r), which adds the two circular bases. For a cylinder with radius 3 and height 10: • Volume: V = π × 3² × 10 = π × 9 × 10 = 282.743 cubic units • Lateral SA: L = 2π × 3 × 10 = 188.496 square units • Base area: B = π × 3² = 28.274 square units • Total SA: T = 188.496 + 2 × 28.274 = 245.044 square units Memory aids: • Volume = base area × height = πr²h • Lateral SA = circumference × height = 2πr × h (imagine unrolling the curved surface into a rectangle) • Total SA = lateral + top + bottom = 2πrh + 2πr² • This calculator works from any two known values — radius, height, volume, or surface area

How do you find the radius or height from the volume?

To find radius from volume and height: rearrange V = πr²h → r = √(V/πh). To find height from volume and radius: rearrange V = πr²h → h = V/(πr²). Examples: • V=100, h=8: r = √(100/(π×8)) = √(3.9789) = 1.9947 units • V=100, r=3: h = 100/(π×9) = 100/28.274 = 3.5368 units For lateral SA: • LSA=188.496, h=10: r = LSA/(2πh) = 188.496/(2π×10) = 188.496/62.832 = 3 units For total SA: • TSA = 2πrh + 2πr² → h = (TSA − 2πr²)/(2πr) — just rearrange linearly in h • TSA=245.044, r=3: h = (245.044 − 2π×9)/(2π×3) = (245.044−56.549)/18.850 = 10 ✓

What is the lateral surface area of a cylinder and why is it useful?

The lateral surface area (also called curved surface area) is the area of the curved side of the cylinder, not including the two circular ends. L = 2πrh. It can be visualised by cutting the curved surface vertically and unrolling it — the result is a rectangle with width 2πr (the circumference) and height h. Why lateral SA matters: • Label area: the area of paper needed to label a tin can (no top/bottom) • Painting: cost to paint only the side of a cylindrical tank • Insulation: material needed to wrap the curved surface of a pipe • Heat transfer: the effective radiating area of a cylinder in many engineering contexts Total SA vs Lateral SA: • Total SA includes two circular ends: T = L + 2πr² • Lateral SA is just the curved surface: L = 2πrh • The ratio L/T = h/(h+r) — increases as h grows, approaches 1 for very tall cylinders • For a cube-like cylinder (h=2r): L/T = 2r/(2r+r) = 2/3 (lateral is 2/3 of total)

What is the diagonal of a cylinder?

The diagonal of a cylinder is the longest straight line that fits inside it — from one edge of a base to the opposite edge of the other base, passing through the interior. It is found using the Pythagorean theorem: d = √(D² + h²) = √((2r)² + h²), where D is the diameter and h is the height. For a cylinder with r=3, h=10: • Diagonal = √((2×3)² + 10²) = √(36 + 100) = √136 ≈ 11.662 units Intuition: imagine tipping a cylinder on its side and looking at the rectangle formed by the diameter and height — the diagonal of that rectangle is the longest internal line of the cylinder. Comparisons: • Height h: vertical extent • Diameter 2r: horizontal extent • Diagonal √(4r²+h²): longest extent — always greater than both h and 2r • For a cylinder where h = 2r (like many cans): d = √(4r²+4r²) = 2r√2

How does a cylinder's volume relate to a cone and sphere of the same radius and height?

This is one of the most beautiful results in classical geometry, proved by Archimedes. For shapes with the same radius r and height h=2r (so the sphere fits exactly inside the cylinder): • Cylinder volume: V_cyl = πr²h = πr²(2r) = 2πr³ • Sphere volume: V_sph = (4/3)πr³ • Cone volume: V_cone = (1/3)πr²h = (2/3)πr³ The ratio is: Cone : Sphere : Cylinder = 1 : 2 : 3 • Cone = 1/3 of cylinder • Sphere = 2/3 of cylinder • This means: V_sphere = V_cylinder − V_cone (for h=2r) Archimedes considered this discovery so important he asked for a sphere inscribed in a cylinder to be carved on his tombstone. The Roman general Marcellus honoured this wish after Archimedes died in 212 BC. This calculator shows cylinder volume; combine it with the sphere and cone calculators to verify the 1:2:3 ratio yourself.

Cylinder Calculator — Volume, Surface Area & Diagonal

The Cylinder Calculator computes all properties of a right circular cylinder from any two known measurements. Five input modes: radius + height (most common), volume + height, volume + radius, lateral surface area + height, or total surface area + radius. All results are shown instantly — volume, la...

CYLINDER CALCULATOR

KNOWN VALUES

Radius (r)

Height (h)

RADIUS (r)

HEIGHT (h)

VOLUME

282.743339

TOTAL SA

245.044227

Lateral SA

188.495559

2πrh

Base Area

28.274334

πr² (×1)

Total SA

245.044227

L + 2×Base

Diagonal (longest)11.661904 units

Diameter6 units

V / h (base area)28.274334

h / r ratio3.3333

STEP-BY-STEP

Given: radius r = 3, height h = 10

Volume: V = πr²h = π × 3² × 10 = 282.743339

Lateral SA: L = 2πrh = 2π × 3 × 10 = 188.495559

Base area: B = πr² = π × 3² = 28.274334

Total SA: T = L + 2B = 188.495559 + 2×28.274334 = 245.044227

Diagonal: d = √((2r)²+h²) = √(36+100) = 11.661904

QUICK EXAMPLES

LIVE DIAGRAM

SURFACE AREA BREAKDOWN

Lateral (curved): 188.495676.9%

Top base: 28.274311.5%

Bottom base: 28.274311.5%

EXACT EXPRESSIONS

Volume

π × 3² × 10 = π × 9 × 10

Lateral SA

2π × 3 × 10

Total SA

2π × 3 × (10 + 3)

Created with❤️byeaglecalculator.com

CYLINDER FORMULAS

ALL CYLINDER FORMULAS

NAME	FORMULA	DESCRIPTION
Volume	V = πr²h	Base area times height
Lateral SA	L = 2πrh	Curved surface only — circumference × height
Base area	B = πr²	Area of one circular end
Total SA	T = 2πrh + 2πr² = 2πr(h+r)	Lateral plus both circular bases
Radius from V,h	r = √(V/πh)	Square root of V over πh
Height from V,r	h = V/(πr²)	Divide volume by base area
Radius from LSA,h	r = LSA/(2πh)	Divide lateral SA by 2πh
Height from TSA,r	h = (TSA − 2πr²)/(2πr)	Rearrange total SA formula
Diagonal	d = √((2r)² + h²)	Longest internal line — Pythagorean theorem on diameter and height

HOW TO USE

1
Select your input mode based on which two values you know. Radius + Height is the most common. The other modes let you work backwards from volume or surface area.
2
Enter your two known values. For Total SA + Radius, the height is calculated by rearranging the total SA formula linearly — a negative result means the values are inconsistent.
3
Volume, lateral SA, base area, total SA, and diagonal all appear instantly. The h:r ratio shows whether the cylinder is squat or tall.
4
Check the surface area breakdown panel — it shows the percentage each component contributes to the total. Useful for material optimisation problems.
5
Read the step-by-step solution to see exactly which formula was rearranged and how the unknown value was derived from your inputs.

WORKED EXAMPLE

Example 1 (r=3, h=10): V=π×9×10=282.743, Lateral SA=2π×3×10=188.496, Base=π×9=28.274, Total SA=245.044, Diagonal=√(36+100)=11.662. Example 2 (V=100, h=8): r=√(100/8π)=√3.979=1.995. Example 3 (TSA=245.044, r=3): h=(245.044−2π×9)/(2π×3)=10.

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Last updated: April 29, 2026 · Formula verified by EagleCalculator team · Eagle-eyed accuracy for every calculation.