There are some blocks of the quadrangular pyramid with 10 cm in length of all the sides and some blocks of the triangular pyramid with 10 cm in length of all the sides.
(1) I make a triangular pyramid with 20 cm in length of all the sides by combining these blocks without any gap.
How many blocks are used, respectively?
(2) Find the volume ratio of one block of the quadrangular pyramid of one-side 10 cm and one block of the triangular pyramid of one-side 10 cm.
Answer
(1) Quadrangular pyramid 2 pieces
Triangular pyramid 4 pieces
(2) 2 : 1
Solution
(1) As shown in the figure below, the triangular pyramid is made of combination of four pieces of triangular pyramid whose one of vertex is A, B, C, and D, respectively and two pieces of quadrangular pyramid.
(2) The homothetic ratio of a large triangular pyramid and a small triangular pyramid is 20 : 10 = 2 : 1.
The volume ratio is (2 × 2 × 2) : (1 × 1 × 1) = 8 : 1.
When the volume of the triangular pyramid of one-side 10 cm is set to 1, the volume of the quadrangular pyramid of one-side 10 cm is (8 - 1 × 4) / 2 = 2.
(1) I make a triangular pyramid with 20 cm in length of all the sides by combining these blocks without any gap.
How many blocks are used, respectively?
(2) Find the volume ratio of one block of the quadrangular pyramid of one-side 10 cm and one block of the triangular pyramid of one-side 10 cm.
Answer
(1) Quadrangular pyramid 2 pieces
Triangular pyramid 4 pieces
(2) 2 : 1
Solution
(1) As shown in the figure below, the triangular pyramid is made of combination of four pieces of triangular pyramid whose one of vertex is A, B, C, and D, respectively and two pieces of quadrangular pyramid.
(2) The homothetic ratio of a large triangular pyramid and a small triangular pyramid is 20 : 10 = 2 : 1.
The volume ratio is (2 × 2 × 2) : (1 × 1 × 1) = 8 : 1.
When the volume of the triangular pyramid of one-side 10 cm is set to 1, the volume of the quadrangular pyramid of one-side 10 cm is (8 - 1 × 4) / 2 = 2.
