Question

tori is creating an informal argument for the formula for the volume of a square pyramid. to start, she considered a cube with side lengths of 10 units. then, tori sliced through the center of the cube along its diagonals, creating identical square pyramids. toris diagram is shown below. how many square pyramids did tori slice the cube into? what is the height of each square pyramid? units pick all the expressions that can be used to represent the volume of the square pyramid. $$ \frac { 1 } { 5 } ( 10 \cdot 10 \cdot 10 ) $$ $$ \frac { 1 } { 6 } ( 10 \cdot 10 \cdot 10 ) $$ $$ \frac { 4 } { 3 } ( 10 \cdot 10 \cdot 10 ) $$ $$ \frac { 1 } { 3 } ( 10 \cdot 10 ) ( 5 ) $$ $$ \frac { 8 } { 3 } ( 10 \cdot 10 ) ( 5 ) $$ $$ \frac { 1 } { 2 } ( 10 \cdot 10 ) ( 6 ) $$

Explanation:

Step1: Determine the number of pyramids

A cube has 6 faces. When slicing through the center of the cube along its diagonals, each face of the cube forms the base of a square - pyramid. So, the number of square - pyramids is 6.

Step2: Calculate the height of each pyramid

The side - length of the cube \(s = 10\) units. The height \(h\) of each square - pyramid is half of the side - length of the cube. So, \(h=\frac{10}{2}=5\) units.

Step3: Calculate the volume of the cube

The volume of the cube \(V_{cube}=s\times s\times s=10\times10\times10\) cubic units.
Since the cube is divided into 6 equal - volume square - pyramids, the volume of each square - pyramid \(V=\frac{1}{6}(10\times10\times10)\).
Also, using the formula for the volume of a pyramid \(V=\frac{1}{3}Bh\), where \(B = 10\times10\) (the area of the base) and \(h = 5\) (the height), we get \(V=\frac{1}{3}(10\times10)\times5\)

Answer:

• The number of square - pyramids: 6
• The height of each square - pyramid: 5 units
• The expressions for the volume of the square - pyramid: \(\frac{1}{6}(10\cdot10\cdot10)\) and \(\frac{1}{3}(10\cdot10)(5)\)