Similarity of bodies
Similar bodies. Mirror similar bodies. Relations of
corresponding sizes in similar figures and bodies.
Two bodies are similar, if one of them can be received from another by increasing (or decreasing) of all its linear dimensions by the same
ratio. A car and its model are similar bodies. Two bodies (or figures) are mirror similar, if one of them is similar to a mirror image of another. For instance,
a picture and its photo negative are mirror similar one to another.
In similar and mirror similar figures all corresponding angles ( linear and dihedral ones) are equal. In similar bodies polyhedral
and solid angles are equal; in mirror similar bodies they are mirror equal.If two tetrahedrons ( two triangle pyramids ) have
correspondingly proportional edges ( or correspondingly similar faces ), then they are similar or mirror similar. For instance, if edges
of the first of them are two times more than the second ones, then also heights, apothems, radius of circumscribed circle of the first pyramid are two times
more than the second ones. This theorem does not take place for polyhedrons with greater number of faces. Assume, that we joined all edges of a cube
in its vertices by hinges; then we can change a shape of this figure without stretching its edges and receive a parallelepiped from an initial cube.
Two regular prisms or pyramids with the same number of faces are similar, if radii of their bases are proportional to their heights. Two circular
cylinders or cones are similar, if radii of their bases are proportional to their heights.
If two or more bodies are similar, then areas of all corresponding plane and curved surfaces of these bodies are proportional to squares of any
If two or more bodies are similar, then their volumes and also volumes of any their corresponding
parts are proportional to cubes
of any corresponding segments.
E x a m p l e . A cup with a diameter 8 cm an a height 10 cm contains 0.5 liter of water.
What sizes of a similar cup, containing 4 liter of water ?
|S o l u t i o n .
|| As cups are similar cylinders, therefore a ratio of their volumes is equal to a
ratio of cubes of corresponding segments ( in our case – heights and diameters
of cups). Hence, a height h of a new cup is found from a ratio:
( h / 10 ) ³ = 4 / 0.5 , that is h ³ = 8
· 10 ³ , hence h = 20 cm;
analogously, for diameter d we’ll receive:
( d / 8 ) ³
= 4 / 0.5 , that is d ³ = 8 · 8
³ , hence d = 16