Cambridge International Mathematics

Mensuration (solids and containers) (Chapter 11) 241

Solids of uniform cross-section

Notice in the triangular prism alongside, that vertical

slices parallel to the front triangular face will all be the

same size and shape as that face. We say that solids

like this are solids ofuniform cross-section. The cross-

Another example is the hexagonal prism shown opposite.

For any solid of uniform cross-section:

Volume=area of cross-section£length

In particular, for acylinder, the cross-section is a circle and so:

Volume=area of circle£length

=¼r^2 £l

i.e., V =¼r^2 l or V=¼r^2 h

PYRAMIDS AND CONES

Thesetapered solidshave a flat base and come to a point called theapex. Theydo nothave identical

cross-sections. The cross-sections always have the same shape, but not the same size.

For example,

Volume=^13 (area of base£height)

A formal proof of this formula is beyond the scope of this course. It may be demonstrated using water

displacement. Compare tapered solids with solids of uniform cross-section with identical bases and the same

heights.

For example: ² a cone and a cylinder

² a square-based pyramid and a square-based prism.

SPHERES

The Greek philosopherArchimedeswas born in Syracuse in 287 BC. Amongst many other important

discoveries, he found that the volume of a sphere is equal to two thirds of the volume of the smallest

cylinder which encloses it.

solid cross-section

lhor

r end

length

cone

h

square-based pyramid triangular-based pyramid

h h

section in this case is a triangle.

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