Babies lose heat faster than adults: why? Stones used to retain or conserve heat in solar traps are large and round while radiators, designed to 'lose' or radiate heat, are rectangular and almost flat: why? The answer to these questions lies in the relationship between the volume of an object and its surface area and the influence on this relationship of size and shape.
To investigate we first set up a worksheet to record different sizes of cube using the formulae shown in the first table. The second table shows the data produced when the formulae are copied into the rows underneath.
Up to an edge length of 5 the surface area is greater than the volume, for a length of 6 the surface area and volume are equal and thereafter the surface area is less than the volume.
The greater the surface area in relation to volume the more heat a body will lose because there is a greater surface to radiate it in relation to the heat source itself. Column E shows how the ratio falls below zero as the volume exceeds surface area, so a smaller value shows more efficient.
For a given volume of material, what dimensions can we compute for, say, a flat-panel radiator? For an edge length of 2, a surface area of 24 and a volume of 8 (as in the table above), what dimensions could this cube take if it was flattened into a radiator shape?
Let us assume that the width of the radiator is 5 units wide and 5 units long so the area of one face is 25 square units. Thus [25 * depth = 8] or [depth = 8/25] = 0.32.
The surface area of this radiator object is the sum of the two faces plus the sum of the sides:
(5 * 5 * 2) + (4 * 5 * 0.32) = 50 + 6.4 = 56.4, compared to the surface area of the original cube of 24.
We can see how flattening the cube increases the surface area in relation to the volume. The ratio SA:V for the cube was 24/8=3 while for the 'radiator' object it is 56.4/8=7.05, a much higher value. If heated the 'radiator' object will radiate more heat than the cube.
A radiator, therefore, is deliberately made rectangular and flat, some with additional panels to transfer as much heat as possible. On the other hand, some houses have large round stones in a glass enclosure to capture heat during the day and radiate it slowly during the night.
To compare the relationship between volume and surface area of a cube with those of a sphere we need to find the dimensions of a sphere which has the same volume as any given cube. We cannot compare a cube of edge length 1 with a sphere of radius or diameter 1, we need to compare objects of the same volume (or surface area). To do this we need to find the radius of the sphere which will have the same area as the cube in question. For example, given a cube of volume 1, what is the radius of a sphere with the same volume? To solve this problem we can use algebra or Goal Seek in Excel.
The formula for the volume of a sphere is 4/3πr3 and for the surface area it is 4πr2.
Set up the following spreadsheet to calculate the volume of a sphere with a radius of 1.
Now select Tools/Goal Seek, set the desired value in cell C4 to 1 and select A4 as the cell to vary. Click OK and the answer 0.62... appears in A4, which is the radius of a sphere of volume 1.
Similarly we can calculate the radius for spheres with volumes equal to that of the other basic cubes, 8, 27, 64, 125, etc. Following this we work out the surface area and volume of spheres with radius values computed from Goal Seek model.
Returning to the main worksheet we enter the radius information into cells A14:A17. We have PI in C1 so the formula in B14 for the surface area is:
=4 * $C$1 * A14 ^ 2 (4πr2)
(Spaces added for clarity.) The formula for the volume of the sphere is:
= 4 / 3 * $C$1 * A14 ^ 3 (4/3πr3)
Now we can see that a cube of volume 1 has a SA:V ratio of 6 while a sphere of the same volume has a ratio of 4.83. Similarly, a cube of volume 8 has a SA:V ratio of 3 while that for a sphere of the same volume is 2.42. We can see, therefore, that, volume for volume, a sphere is a better shape for conserving energy than a cube because it has a better ratio between surface area and volume.
A baby has smaller limbs than an adult and so loses heat (by convection) more quickly, making it much more vulnerable to cold. A blanket or warm coat traps warm air around the skin and prevents convection taking it away. The size of a human head balances the heat of the blood which flows through the brain with the need to lose heat and maintain the correct temperature. A larger head would retain more heat and result in overheating - we would need large ears or some other mechanism to lose more heat! People living near the poles have rounder heads (to conserve heat) than those in the tropics who have a more elongated head (to aid convection and stay cool). The relationship between surface area and volume affects heat loss from bodies ranging from beakers of water to planets and stars.
Use the Back Button to Return to the previous page