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Example (1): Surface area to volume ratio of simply connected 3 dimensional geometric objects.

A simply connected object in space is the one where every two points A and B in the object can be connected by a line segment AB which is entirely inside the object. Following solid objects are all simply connected objects.

Surface area to volume relationship of a geometric object plays an important role in everyday life and events. Each object in space divides the space into two regions: interior of the object and surrounding region. Surface of the object (which is part of the object) acts as the boundary. Through the surface, the object and its surrounding can exchange force, materials, energy and son on.  And exchange between the object and the surrounding can be called flow such as flow of energy. Magnitude of such flow could be directly proportional to the boundary surface area.

We choose three simple objects (see the diagram below) with a principal dimension indicated by labels x, y, and z. Their surface areas A and volumes V are formulated in terms of the principal dimension. For some selected values of principal dimension, A and V values are calculated using spread sheet for all three objects. Then, plots of A against V are put together in the same graph.

Observation:

• For each object, the smaller the volume the greater the A/V ratio.

• Among the three selected, right circular cylinder has smallest A/V ratio for all volumes.

Possible application:

If the flow through the surface is thermal energy, then amount of thermal energy excchange per one unit volume may be an important factor. Thermal energy flow is  directly proportional to the surface area. That is, the greater the surface area, the greater the thermal energy flow. Smaller A/V ratio shall be preferable as a factor if the reduction of thermal energy flow is necessary. Greater A/V ratio shall be preferable as a factor if maximizing the flow of thermal energy is necessary as in the case of heat exchangers.

Extension:

Interested person can do the same or similar calculations for different simple objects such as sphere and right circular cylinder capped with hemispheres.

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