Right Cylinder Calculator: Volume, Surface Area, Lateral Area
Enter the radius and height of a right circular cylinder to get its volume (V), total surface area (A), lateral surface area (A_l), base area (A_b), space diagonal, and surface-to-volume ratio. Choose from seven length units, or switch to one of six solve modes to reverse-calculate the radius or height from a known volume, lateral area, or total surface area. Every result comes with a live step-by-step breakdown.
Formula
Worked example
Cylinder with r = 5 cm, h = 10 cm: C = 2 x pi x 5 = 31.42 cm; A_b = pi x 25 = 78.54 cm^2; A_l = 2 x pi x 5 x 10 = 314.16 cm^2; A = 2 x pi x 5 x 15 = 471.24 cm^2; V = pi x 25 x 10 = 785.40 cm^3; d = sqrt(100 + 100) = 14.14 cm.
What is a right cylinder?
A right circular cylinder is a three-dimensional solid bounded by two parallel congruent circles (the bases) and a curved lateral surface connecting them. "Right" means the axis joining the centres of the two bases is perpendicular to both bases, as opposed to an oblique cylinder where the axis is tilted. Almost every real-world cylinder you encounter - food cans, pipes, columns, tanks - is a right circular cylinder. The geometry is controlled by just two measurements: the base radius r and the height h.
How the four key measurements relate to each other
Volume, lateral area, total area, and base area are all simple functions of r and h. Because of this, knowing any two of the five main quantities (r, h, V, A_l, A) is enough to determine the rest by algebra. For example, if you know the volume and the radius, height follows from h = V / (pi x r^2). If you know the lateral area and the height, radius follows from r = A_l / (2 x pi x h). This calculator covers all six pairwise solve modes so you never need to rearrange the formulas yourself.
Lateral area versus total surface area
The lateral surface area (A_l = 2 x pi x r x h) covers only the curved side wall. If you cut along a vertical line and unrolled it, you would get a rectangle with width equal to the base circumference (2 x pi x r) and height equal to h. The total surface area (A = A_l + 2 x A_b = 2 x pi x r x (r + h)) adds the two circular end caps. Use lateral area when you need to cover the side only - for example, labelling or wrapping a can. Use total area when the entire surface must be painted, plated, or insulated.
The optimal can: minimising material for a fixed volume
Calculus shows that, for a fixed volume, the right cylinder with the smallest total surface area has h = 2r (height equals diameter). At that proportion the surface-to-volume ratio A/V = 2(r + h) / (r x h) reaches its minimum of 3/r. Most commercial beverage cans are close to this ratio, though slight deviations appear because of manufacturing constraints and stackability. You can experiment in this calculator by fixing the volume (use the h_V or r_V mode) and varying the dimensions to watch the total area change.
Right cylinder formulas at a glance
| Quantity | Formula | Notes |
|---|---|---|
| Volume (V) | V = pi x r^2 x h | Cubic units |
| Lateral area (A_l) | A_l = 2 x pi x r x h | Unrolls to a rectangle (2pi*r) x h |
| Base area (A_b) | A_b = pi x r^2 | One circular end cap |
| Total area (A) | A = 2 x pi x r x (r + h) | A_l + 2 A_b |
| Circumference (C) | C = 2 x pi x r | Perimeter of each base |
| Space diagonal (d) | d = sqrt(4r^2 + h^2) | Longest line through interior |
| SV ratio | A/V = 2(r + h) / (r x h) | Minimum at h = 2r |
| Optimal h | h = 2r (min surface) | Classic tin-can result |
All formulas use radius r and height h of the cylinder.
Frequently asked questions
What is the formula for the volume of a right cylinder?
V = pi x r^2 x h, where r is the base radius and h is the height. For a cylinder with radius 5 cm and height 10 cm, V = pi x 25 x 10 = 785.40 cm^3 (to 2 decimal places). You can read this as the base area (pi x r^2) multiplied by the height, which generalises to any prism or cylinder shape.
What is lateral surface area and how is it different from total surface area?
Lateral surface area (A_l = 2 x pi x r x h) is the area of the curved side wall only. Total surface area (A = 2 x pi x r x (r + h)) adds the area of both circular bases (2 x pi x r^2). Use lateral area when you only need to cover the side (labelling, wrapping), and total area when you need to enclose the cylinder completely (painting, coating).
How do I find the radius or height if I know the volume and one dimension?
Rearrange V = pi x r^2 x h. To find height given radius and volume: h = V / (pi x r^2). To find radius given height and volume: r = sqrt(V / (pi x h)). In this calculator, select the matching solve mode from the "Solve for" dropdown and enter your two known values.
What is the space diagonal of a cylinder?
The space diagonal (d) is the longest straight line that fits inside the cylinder. It connects a point on the rim of one base to the diametrically opposite point on the other base, passing through the interior. Its length is d = sqrt((2r)^2 + h^2) = sqrt(4r^2 + h^2). This is analogous to the space diagonal of a rectangular box.
Why does the optimal can have height equal to its diameter?
For a fixed volume V, the total surface area A = 2 x pi x r x (r + h) is minimised when dA/dr = 0 (treating h as a function of r via V = pi x r^2 x h). Differentiating and setting equal to zero gives the condition h = 2r. At this proportion the surface-to-volume ratio equals 6 / (2r) = 3/r, the theoretical minimum for a right cylinder.