Areas and Volumes Problems

1) DABC is equilateral.

X is on AB and AX:XB=1:2.

Y is on BC and BY:YC=1:2.

Z is on CA and CS:ZA=1:2.

AY, BZ, and CX intersect at P,Q,R.

Prove that the area of DPQR is one-seventh of the area of DABC.

2) Find the diagonal of a cube with total surface area of 1296 sq units.

3) Find the volume of a regular octahedron (all faces are equilateral).

If:

a) each edge is 6.

b) space diagonal is 6.

4) A regular hexagon pyramid has a face which makes an 80^o angle with its base. If the base edge is 12, find:

a) the lateral surface area (without base)

b) the total surface area (including base)

5) Given a regular hexagonal pyramid with a base perimeter of 72 and an altitude of , find the total surface area and volume.