The four diagonals of a cube are drawn to create 6 square pyramids with the same base and height. The volume of the cube is $(b)(b)(b)$. The height of each pyramid is h.
Therefore, the volume of one pyramid must equal one-sixth the volume of the cube, or
A. $\frac{1}{6}(b)(b)(2 h)$ or $\frac{1}{3} B h$.
B. $\frac{1}{6}(b)(b)(6 h)$ or $B h$.
C. $\frac{1}{3}(b)(b)(b h)$ or $\frac{1}{3} Bh$.
D. $\frac{1}{3}(b)(b)(2 h)$ or $\frac{2}{3} B h$.
Answer (1)
The volume of the cube is b 3 .
The cube is divided into 6 congruent pyramids, so the volume of each pyramid is 6 1 b 3 .
The volume of a pyramid is 3 1 B h , where B = b 2 is the base area and h is the height.
The correct expression for the volume is 6 1 ( b ) ( b ) ( 2 h ) or 3 1 B h , where h = 2 1 b .
3 1 B h
Explanation
Problem Analysis The problem states that a cube is divided into 6 congruent square pyramids by drawing the four diagonals of the cube. The volume of the cube is given as b × b × b = b 3 . We need to find the volume of one of these pyramids.
Volume of One Pyramid Since the cube is divided into 6 equal pyramids, the volume of one pyramid is one-sixth of the volume of the cube. Therefore, the volume of one pyramid is: 6 1 × b 3 = 6 1 b 3
Pyramid Volume Formula The volume of a pyramid is also given by the formula 3 1 B h , where B is the area of the base and h is the height of the pyramid. In this case, the base is a square with side length b , so the area of the base is B = b 2 . Thus, the volume of the pyramid can be expressed as 3 1 b 2 h .
Equating Volumes We know that the volume of the pyramid is 6 1 b 3 . We also have the expression 3 1 b 2 h for the volume. Equating these two expressions, we get: 6 1 b 3 = 3 1 b 2 h
Solving for Height Solving for h , we have: h = 3 1 b 2 6 1 b 3 = 6 1 × 1 3 × b 2 b 3 = 6 3 b = 2 1 b So, h = 2 1 b .
Confirming the Volume Now we can express the volume of the pyramid as 3 1 B h = 3 1 b 2 h . Substituting h = 2 1 b , we get: 3 1 b 2 ( 2 1 b ) = 6 1 b 3 This confirms that the volume of the pyramid is indeed 6 1 b 3 .
Analyzing the Options Now, let's examine the given options:
6 1 ( b ) ( b ) ( 2 h ) = 3 1 b 2 h = 3 1 B h . Since h = 2 1 b , this is 3 1 b 2 ( 2 1 b ) = 6 1 b 3 . This is the correct volume.
6 1 ( b ) ( b ) ( 6 h ) = b 2 h = B h . Since h = 2 1 b , this is b 2 ( 2 1 b ) = 2 1 b 3 . This is incorrect.
3 1 ( b ) ( b ) ( bh ) = 3 1 b 3 h . This expression is dimensionally incorrect, as it involves h multiplied by b , which is not what we want.
3 1 ( b ) ( b ) ( 2 h ) = 3 2 b 2 h = 3 2 B h . Since h = 2 1 b , this is 3 2 b 2 ( 2 1 b ) = 3 1 b 3 . This is incorrect.
Final Answer The correct expression for the volume of one pyramid is 6 1 ( b ) ( b ) ( 2 h ) , which simplifies to 3 1 B h .
Examples
Understanding the volume of pyramids derived from a cube has practical applications in architecture and engineering. For example, consider designing a modern art installation consisting of interconnected pyramids. Knowing the precise volume of each pyramid, based on the dimensions of the cube from which they are conceptually derived, allows for accurate material estimation and structural planning. This ensures the installation is both aesthetically pleasing and structurally sound, demonstrating a blend of artistic vision and mathematical precision.
