Calculate Multiple Area and Volume

Calculator for the lengths at a multiplication of area or volume and vice versa. If an area is doubled (factor=2), then its edge length multiplies by square root of two, at a triplication, it multiplies by square root of three. If a volume is doubled, then its edge length multiplies by third root of two, at a triplication, it multiplies by third root of three. And so on. This is not only for the lengths of straight objects like square or cube, but also e.g. for the radius of circle and sphere.

l = length, d = dimension, f = factor area, volume

l = ^d√ f f = l^d d = log_l(f)

The doubling of the area of a square to a square twice the area and of the volume of a cube to a cube twice the volume:
Doubling Area, Volume

The doubling of an area means the multiplication of a length by √ 2 = 1.414213562373095.
The doubling of a volume means the multiplication of a length by ³√ 2 = 1.259921049894873.

Example: a cylindrical glass with 8 cm diameter and 10 cm height can be filled with half a liter. To have a glass with one liter, diameter and height have to be multiplied by ³√2, so a glass with approximately 10 cm diameter and 12.6 cm height is needed.

We are three-dimensional beings and everything around us is three-dimensional. Nevertheless, we are not used to correctly estimating volumes when they change size. The content of large things is often underestimated. It is also counterintuitive that a change in size by a factor of 1.26 is enough to double the volume.
Another source of error is comparing the contents of compact objects with those that are longer in one direction. Tall, thin glass gives the impression that more can fit into it than into one where the length, width and height are roughly the same. In fact, the surface area of long, thin objects tends to be higher than that of compact ones, and perhaps this fact contributes to the misjudgment. However, misjudgments are best avoided if you are aware that and when they can occur.

Length:
Dimension:
Factor area, volume: