4. Geometric Modeling

Lesson

You already have seen one explanation for understanding the volume of a pyramid. Now, you'll investigate another. After exploring each applet, answer the questions that follow.

Yangma is an ancient Chinese name for a rectangular-based pyramid whose vertex is in line with one of the vertices of the base. In the applet below you'll see a cube dissected into three yangma.

1. Move the slider on the right to see the three yangma. Do all three yangma have the same volume? Explain your answer.

2. Recall that the volume of a cube $V=s^3$`V`=`s`3 , where $s$`s` stands for the length of an edge on the cube. Express the volume of one yangma using the volume of a cube.

3. Now, change the cube to a prism by dragging the point on the cube. Do you think that all three yangma still have the same volume? Explain your answer.

From the yangma, you saw how a right square pyramid is $\frac{1}{3}$13 the volume of a cube (or prism of equal height). Can the same be said of all pyramids (not just right ones)? As you explore the applet below, think about how Cavalieri's Principle relates. Then, answer the questions.

1. Move point $J$`J` up and down and compare the cross section areas ($A_1$`A`1 and $A_2$`A`2). What do you notice?

2. Move the vertex of one pyramid by moving the blue point. Compare the volumes of the two pyramids ($V_1$`V`1 and $V_2$`V`2). What do you notice?

3. Summarize what you see here. Note how it relates to Cavalieri's Principle and the formula for the volume of all pyramids.

A Greek philosopher and mathematician by the name of Archimedes is known, among other things, for calculating areas and volumes by breaking objects into infinitesimally small pieces (that's really really small). You'll learn more about this when you study calculus, but for now, you can see an application in the applet below.

1. What figure does the pyramid start to resemble as the number of faces increases?

2. Recall the formula for the volume of a cone. What similarities does it have to the formula for the volume of a pyramid?

2. After looking at the applet below, explain the connection between the two formulas.

Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone. Use dissection arguments, cavalieri's principle, and informal limit arguments.