# Answer to Question #9432 in Geometry for John

Question #9432

how many symmetry planes does a rectangular solid have?

Expert's answer

By "rectangular solid" we will mean a "rectangular parallelepiped".

So

this is a polyhedron Q with 6 faces each beibng a ractangular, and each pair of

oposite faces are congruent.

Denote its vertexes by

A,B,C,D, A', B',

C', D'

So that there are 3 pairs of equal faces:

ABCD =

A'B'C'D'

ADD'A' = BCC'B'

ABB'A' = DCC'D'

Then the following

three planes are the symmetry planes of Q:

1) let p1 be the plane passing

through the middle point of AA' and orthogonal to AA'.

2) let p2 be the

plane passing through the middle point of AB and orthogonal to AB.

3) let

p3 be the plane passing through the middle point of AD and orthogonal to

AD.

If all three lengths AA', AB, AD are distinct, then p1, p2, and

p3 are all the symmetry planes.

If, say, AA'=AB and these sides

differs from AD, then there are additional 2 symmetry planes

p4 =

AB'C'D

p5 = A'BCD'

Finally, if all sides are the same AA'=AB=AD,

so Q is a cube, then there are also additional 4 planes:

p6 =

ABC'D'

p7 = A'B'CD

p8 = AA'C'C

p9 = BB'D'D

So

this is a polyhedron Q with 6 faces each beibng a ractangular, and each pair of

oposite faces are congruent.

Denote its vertexes by

A,B,C,D, A', B',

C', D'

So that there are 3 pairs of equal faces:

ABCD =

A'B'C'D'

ADD'A' = BCC'B'

ABB'A' = DCC'D'

Then the following

three planes are the symmetry planes of Q:

1) let p1 be the plane passing

through the middle point of AA' and orthogonal to AA'.

2) let p2 be the

plane passing through the middle point of AB and orthogonal to AB.

3) let

p3 be the plane passing through the middle point of AD and orthogonal to

AD.

If all three lengths AA', AB, AD are distinct, then p1, p2, and

p3 are all the symmetry planes.

If, say, AA'=AB and these sides

differs from AD, then there are additional 2 symmetry planes

p4 =

AB'C'D

p5 = A'BCD'

Finally, if all sides are the same AA'=AB=AD,

so Q is a cube, then there are also additional 4 planes:

p6 =

ABC'D'

p7 = A'B'CD

p8 = AA'C'C

p9 = BB'D'D

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