# Similarity of bodies

*Similar bodies. Mirror similar bodies. Relations of*

corresponding sizes in similar figures and bodies.

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

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
*

*corresponding segments.*

*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;d we’ll receive:d / 8 ) ³= 4 / 0.5 , that is d ³ = 8 · 8³ , hence d = 16cm . |