Length, Area and Volume

Length is a measure of distance. In the International System of Units (SI) system the base unit for length is the metre.

Length is commonly understood to mean the most extended dimension of a fixed object. However, this is not always the case and may depend on the position the object is in.

Various terms for the length of a fixed object are used, and these include height, which is vertical length or vertical extent, and widthbreadth or depth. Height is used when there is a base from which vertical measurements can be taken. Width or breadth usually refer to a shorter dimension when length is the longest one. Depth is used for the third dimension of a three dimensional object.

There are times when it is desired to measure things much smaller or much bigger than the base unit of measurement. It is convenient to use prefixes to form decimal multiples and submultiples of the SI units for these measurements.

The common prefixes are:

Most of use will only use a subset of these.

When doing calculations convert measurements to the same units.

Length is the measure of one spatial dimension, whereas area is a measure of two dimensions (length squared) and volume is a measure of three dimensions (length cubed).

A square measuring 1m x 1m would measure 1m2 in area.

There are for example 100cm in a metre.

An area of one square metre would be the same as 100cm x 100cm = 10,000cm2.

Area = length by width

A volume of one cubic metre would be the same as 100cm x 100cm x100cm = 1,000,000cm3 = 106 cm3.

Volume = length x width x height = area of base x height.

Another unite, the litre, is most commonly used for items (such as fluids and solids that can be poured) which are measured by the capacity or size of their container, whereas cubic metres (and derived units) are most commonly used for items measured either by their dimensions or their displacements.

A litre is a cubic decimetre, which is the volume of a cube 10 centimetres × 10 centimetres × 10 centimetres (1 L ≡ 1 dm3 ≡ 1000 cm3). Hence 1 L ≡ 0.001 m3 ≡ 1000 cm3, and 1 m3 (i.e. a cubic metre, which is the SI unit for volume) is exactly 1000 L.

There are some shapes for which we may often require measures of surface areas or volumes. These will be collected together in Formulae and Tables booklets so it is not necessary to remember them all.

A prism is a solid object with two identical ends and flat sides. The shape of the ends give the prism a name.

• The cross section is the same all along its length
• The sides are parallelograms (4-sided shape with opposites sides parallel)
• It is also a polyhedron

In a regular prism, the base of the prism is in the shape of a regular polygon.

In an irregular prism, the base of the prism is in the shape of an irregular polygon.

A pyramid is a solid object where:

• The sides are triangles which meet at the top (the apex).
• The base is a polygon (a flat shape with straight sides)

• It is also a polyhedron.

The shape of the base gives the pyramid a name.

When the base is a regular polygon it is a Regular Pyramid, otherwise it is an Irregular Pyramid.

When the apex is directly above the centre of the base it is a Right Pyramid, otherwise it is an Oblique Pyramid.

Strictly speaking a cylinder is not a prism and a cone is not a pyramid, because their bases are not polygons, however they are extremely similar. If you imagine a prism with regular polygons for bases, as you increase the number of sides, the solid gets to look just like a cylinder. So we can say that a cylinder is a prism with an infinite number of faces. Similarly with a cone as a pyramid.

frustum, (plural: frusta or frustums) is the portion of a solid (normally a cone or pyramid) that lies between one or two parallel planes cutting it. A right frustum is a parallel truncation of a right pyramid or right cone

If all the edges are forced to be identical, a frustum becomes a uniform prism.

Cones and pyramids can be viewed as degenerate cases of frusta, where one of the cutting planes passes through the apex (so that the corresponding base reduces to a point).