Pyramids: where the apex projects
From equal lateral edges or equal dihedral angles, decide which centre of the base the apex projects to, then find the height and the lateral area.
- Ukraine: grade 11
- Russia: grade 10
Guide
In the Ukrainian programme this is Grade 11 geometry, §18 "Pyramid". The projection of the apex on the plane of the base is the foot of the height. The textbook proves three statements about this point.
1) If the lateral edges are equal (or make equal angles with the base plane), the apex projects to the circumcentre of the base polygon. For a right-angled triangle base it is the midpoint of the hypotenuse.
2) If all the dihedral angles at the base edges of a convex pyramid are equal, the apex projects to the incentre of the base polygon. For a rhombus it is the intersection of the diagonals.
3) If every dihedral angle at the base edges is α, the lateral area is Sб = Sосн/cosα. All lengths are in centimetres.