3-D Geometry

STEP 1999, Math III, #4: A polyhedron is a solid bounded by

plane faces, which meet in edges and vertices.

You may assume Euler's formula, that

.

In a regular polyhedron the faces are equal regular

-sided polygons, of which meet at each vertex. Show that

where

.

[Hint: You will need to find a relationship between V, m, and n then another relationship between E, m, and n and then sub these into Euler's formula.]

By considering the possible values of

, or otherwise, prove that there are only five regular polyhedra, and find , , and for each.

STEP 2001, Math II, #3: The cuboid ABCDEFGH is such that AE, BF, CG, DH are perpendicular to the opposite faces ABCD and EFGH,

and AB = 2, BC = 1, AE =

. Show that if is the acute angle between the diagonals AG and BH then

be the ratio of the volume of the cuboid to its surface area. Show that for all possible values of .

Prove that, if then .

STEP 2002, Math I, #6: A pyramid stands on horizontal ground. Its base is an equilateral triangle with sides of length

, the other

three sides of the pyramid are of length

and its volume is . Given that the formula for the volume of a pyramid is

X area of base X height ,

show that

The pyramid is then placed so that a non-equilateral face lies on the ground. Show that the new height,

, of the pyramid is given by

, , and . Then use that height to get the area of the new base.]

STEP 2010, Math II, #6: Each edge of tetrahedron ABCD has unit length. The face ABC is horizontal, and P is the point in ABC that is vertically below D.

(i) Find the length of PD.

(ii) Show that the cosine of the angle between adjacent faces of the tetrahedron is 1/3.

(Hint: The angle formed by two intersecting planes is defined to be the angle formed by their normal vectors.)

(iii) Find the radius of the largest sphere that can fit inside the tetrahedron.

STEP 2007, Math I, #5: Note: a regular octohedron is a polyhedron with eight faces each of which is an equilateral triangle.

(i) Show that the angle between any two faces of a regular octohedron is

, height and slant height . Its volume , and the area of its curved surface, are given by

is fixed and is chosen so that is at its stationary value, show that and that .

(ii) Given, instead, that

and respectively (both are acute.)

The point C lies on the circular path with DC perpendicular to AB and the angle of elevation of the top of the flagpole from C is

.

Show that

.

Show that, for any

and ,

and are positive and , then

and hence show that when .