Q1. - Geometry

Question

The diagonals of a cube intersect to divide the cube into six congruent pyramids. That base of each pyramid is a face of the cube, and height of each pyramid is $\frac{1}{2}e$.

1. Use the formula for the volume of a cube to explain why the volume of each pyramid is $V=\frac{1}{6}{e}^3$.
2. Use the formula in part (a) to show that $V=\frac{1}{3}Bh$.

Step-by-Step Solution

Answer

The proof is shown in the below mentioned steps.

a. Step 1. Given information.

The diagonal of a cube intersects to divide the cube into six congruent pyramids as shown. The base of each pyramid is a face of the cube, and height of each pyramid is $\frac{1}{2}e$.

Step 2. Write the concept.

The edge length of cube is equal to $e$. The volume of cube is $e^3$.

Step 3. Determine the volume.

It is given that it is divided into six congruent pyramids. All of the six pyramids are similar to each other. So,

                                                     

 $\begin{array}{l}6\left(\text{Volume\hspace{0.33em}of\hspace{0.33em}pyramid}\right)=e^3\\ \text{Volume\hspace{0.33em}of\hspace{0.33em}pyramid}=\frac{1}{6}{e}^3\end{array}$

Therefore, the volume of each cube is equal to $V=\frac{1}{6}{e}^3$.

b. Step 1. Given information.

The diagonal of a cube intersects to divide the cube into six congruent pyramids as shown. The base of each pyramid is a face of the cube, and height of each pyramid is $\frac{1}{2}e$.

Step 2. Write the concept.

The base of each pyramid is $e$ and height of the pyramid is $\frac{1}{2}e$.

Step 3. Determine the volume.

From part (a), the volume of a pyramid is $V=\frac{1}{6}{e}^3$.

 

Now, calculate the volume of pyramid in terms of base area as:

 $\begin{array}{c}V=\frac{1}{3}\left(e^2\right)\left(\frac{1}{2}e\right)\\ =\frac{1}{3}Bh\end{array}$

Hence, it is proved that the volume of the pyramid is $V=\frac{1}{3}Bh$.