Geometry // Volume Narrative Step 6

Rhombic Prisms

Equivalence through Diagonal Transformation

Material Analysis

Solid rhombic prism, divided rhombic prism (split into two triangular prisms along the major diagonal), major and minor diagonal tickets, ruler.

Presentation: The Transformation

Hold the solid rhombic prism and the divided one side-by-side. Slide the two triangular pieces to form a rectangle.
"Look at our rhombus. If we cut it along this long line—the major diagonal—we get two triangles. If I slide them, what shape do we have now? A rectangle! We haven't changed the volume, only the appearance."
MAJOR D / 2 MINOR d
FIG 6.1: SLIDING TRIANGLES TO FORM RECTANGLE
"In our new rectangle, the base is the minor diagonal ($d$) and the height is half of the major diagonal ($\frac{D}{2}$). To find the volume, we multiply this area by the prism's height ($h$)."
Volume = $(d \times \frac{D}{2}) \times h$
"Place the material on the floor for this one. Having the children look down on the transformation makes the relationship between the diagonals and the rectangular sides much clearer."