As for quadrangular pyramid O-ABCD in a figure, the bottom is a square and all the length of OA, OB, OC and OD is equal. K, L, M, N, P, Q, R, and S are the middle points of each side.
This quadrangular pyramid is cut by three planes of the plane which passes along P, K, N, R and the plane which passes along P, L, M, R and the plane which passes along S, K, L, Q and divides into some solids.
Find the ratio of the volume of a solid including the vertex O and the volume of the quadrangular pyramid O-ABCD.
This quadrangular pyramid is cut by three planes of the plane which passes along P, K, N, R and the plane which passes along P, L, M, R and the plane which passes along S, K, L, Q and divides into some solids.
Find the ratio of the volume of a solid including the vertex O and the volume of the quadrangular pyramid O-ABCD.
Answer
Fig. 2 is a sketch which is cut by the plane which passes along QMNS in addition to the three planes.
As for this quadrangular pyramid O-ABCD, it consists of six blue quadrangular pyramids and four red triangular pyramids in Fig. 2.
Each figure is extracted to Fig. 3.
When being cut by three planes, the solid including the vertex O consists of two quadrangular pyramids same as O-KLMN and one red triangular pyramid.
O-KLMN and O-ABCD are homothetic and a homothetic ratio is
OK : OA = 1 : 2.
When volume of O-ABCD is set to 1, the volume of O-KLMN is 1/2 × 1/2 × 1/2 = 1/8.
Moreover, the volume of one red triangular pyramid is/(1 - 1/8 × 6) / 4 = 1/16.
Therefore, the volume of a solid including the vertex O is 1/8 × 2 + 1/16 = 5/16 and the volume ratio to be found is 5 : 16.
5 : 16
Solution
Fig. 1 is a sketch that quadrangular pyramid O-ABCD is cut by three planes written in the problem.
Fig. 2 is a sketch which is cut by the plane which passes along QMNS in addition to the three planes.
As for this quadrangular pyramid O-ABCD, it consists of six blue quadrangular pyramids and four red triangular pyramids in Fig. 2.
Each figure is extracted to Fig. 3.
When being cut by three planes, the solid including the vertex O consists of two quadrangular pyramids same as O-KLMN and one red triangular pyramid.
O-KLMN and O-ABCD are homothetic and a homothetic ratio is
OK : OA = 1 : 2.
When volume of O-ABCD is set to 1, the volume of O-KLMN is 1/2 × 1/2 × 1/2 = 1/8.
Moreover, the volume of one red triangular pyramid is/(1 - 1/8 × 6) / 4 = 1/16.
Therefore, the volume of a solid including the vertex O is 1/8 × 2 + 1/16 = 5/16 and the volume ratio to be found is 5 : 16.
