# A Mathematical Discussion of Efficiency Ratios

**REVIEWED BY****David Wilson**, Maths lesson resource creator, former maths teacher, and current owner of www.doingmaths.co.uk

Stranded in a snow blizzard with no shelter available, you instinctively assume a foetal position to try to keep warm. Why is this a better body posture than remaining upright with arms and legs outstretched?

An arctic fox does the same thing to keep warm at night.

You might think that it’s simply about minimising surface area, since a foetal position exposes less of your body to the elements. However, surface area is not the only variable that needs to be considered.

In nature and in living creatures, the aim is to minimise energy. This is why water bubbles are spherical. A sphere is the optimal shape that minimises the amount of energy needed to hold water molecules together.

Interaction between the surface of the body and the environment is made easier if the surface area is as large as possible whilst minimising its volume.

The body of the sea sponge is designed to maximise the water that passes through its membranes and deposits food particles at the sponge’s central feeding cavity. This process would not be as efficient if the volume of the sponge were large in relation to its surface area, since the water would have further to travel and food particles may be deposited along the way.

An increased surface area to volume ratio means increased exposure to the environment. The many thin tentacles of a jellyfish increase the surface area, thus allowing more of the surrounding water to be sifted for food.

In humans, a greater surface area allows organs to function more efficiently.

- Our lungs have numerous internal branches that increase the surface area through which oxygen is passed into and carbon dioxide is released from the blood.
- Our intestine has a finely wrinkled internal surface, increasing the area through which nutrients are absorbed by the body.
- The outer layer of the brain is where most thinking activity occurs. With more surface area, there is more capacity for thinking.
- In animals, brains are limited by skull size, and mammal brains have many folds which increase the surface area in a fixed space.

In particular, as an example, children are at a higher risk of hypothermia (loss of body heat), than adults.

This suggests that the ratio of a child’s surface area to their volume is larger in comparison to that of an adult.

Chemically, materials with a high surface to volume ratio react much quicker than when the ratio is lower. Finely ground salt dissolves much quicker than coarse salt.

## Efficiency Ratio

The efficiency ratio (*E*) is the ratio of an object’s surface area (*A*) to its volume (*V*).

The equation is *E* =* A* ÷ *V*, and the units are units of length such as square metre per cubic metre (m^{2}/m^{3}).

We can also express the efficiency ratio as a percentage.

*E* =* 100 *x* A* ÷ *V*

For example, suppose a rock has surface area 88 cm^{2} and volume 48 cm^{3}.

Then its efficiency ratio percentage will be *E* = 100 x 88÷ 48 = 183%.

Efficiency ratios are important in biology with the study of heat loss of animals, in chemical reactions and in physics involving heat loss/gain. The greater the value of *E*, the more efficient is the system under consideration in terms of maximising energy transfer between the object and its environment.

## Efficiency of a Sphere

We will now derive the expression that describes the efficiency ratio of a sphere.

Formulae for the surface area and the volume of a sphere are shown below.

The efficiency ratio for a sphere is

The graph of *E* as a function of *r* has the following shape.

The graph shows that a smaller radius produces a larger efficiency ratio.

A point of reference is to use the value *r* = 3.

When *r* is greater than 3, the efficiency ratio is smaller than 1, which means the surface area of the sphere is smaller than its volume.

This may explain why the giant Pill millipede curls up in a spherical shape when it is threatened.

Its surface area, *A*, is 4×π×3.5^{2} which is approximately 154 cm^{2}.

Its volume, *V*, is 4×π×3.5^{3}÷3. This is approximately 180 cm^{3}.

The ratio of surface area to volume is *A*÷*V*=3÷3.5≈0.86

The graph of *A* against *V* is shown below.

## Our Earth

By taking our Earth to be spherical, its radius is 6371 km, so the percentage efficiency ratio is 100×(3÷6371) which is approximately 0.0047%. This very small efficiency ratio may account for why heat from deep beneath Earth does not reach the surface crust at a depth to be dangerous, since a large unit volume dissipates the heat before it can filter to the smaller surface area.

## Manufacturing

In manufacture, production costs can be reduced by minimising the efficiency ratio.

The surface area of the aluminium can of soft drink 2π*r*(*r* + *h*), where *r* is the radius of the circular base and *h* is the height. Hence, the surface area is 2× 3.14 × 3 × (3 + 14) ≈ 320 cm^{2}.

The volume of the aluminium can, which is cylindrical in shape, is found using the formula π*r*^{2}*h* = 3.14 × 3^{2 }× 14 ≈ 396 cm^{3}. The efficiency ratio is therefore 320 ÷ 396 ≈ 0.8. The small surface area to volume ratio indicates that more drink is contained in less surface area of aluminium. This means less aluminium is required to produce the required amount of drink.

## To the Moon

As a last example, let’s work out the efficiency ratio of the space command module shown below.

The capsule has the shape of a conical frustum.

Its surface area is π[*s*(*R* + *r*) + *R*^{2} + *r*^{2}] = 3.14[3.76(1.95 + 1) + 1.95^{2} + 1^{2}] ≈ 50 m^{2}.

The volume is π*h*[ *R*^{2} + *r*^{2} +*rR*] ÷3 = 3.14 × 3.22 ´ [1.95^{2} + 1^{2} + 1´ 1.95]÷3 ≈ 23 m^{3}.

The efficiency ratio is 50÷23 ≈ 2.2, suggesting surface area is maximised at the expense of interior space. This might be to allow more heat shields to be used to minimise outside temperature on re-entry.

## A Last Word

I have provided an introduction to the significance of Efficiency Ratio in nature and in our day-to-day living. However, by further reading, you will appreciate that Efficiency Ratios are as ubiquitous as Fibonacci’s Golden Ratio.

*This content is accurate and true to the best of the author’s knowledge and is not meant to substitute for formal and individualized advice from a qualified professional.*