Changing dimensions, and the volume of a cone

This lesson asks you to show how changing dimensions affects perimeter, area and volume, and to solve problems with the relationships among volumes. The museum offers the square whose side doubles and the one-third rule for the volume of a cone or pyramid; prisms, cylinders and surface area are for your class to bring. ¹

Draw the square with side one, two and three, and record the areas. Ask how the area changes when the side doubles, then think about what might happen to a volume when all the dimensions double.

In al-Khwarizmi's chapter on mensuration, as the record reports it, the volume of a pointed cone or pyramid is a third of its base times its height, and a truncated one is completed to a point and the added part subtracted ^{2 3}. The study of al-Kashi's muqarnas measurement carries a related lesson on scale. Its Figure 8 gives approximately 0.707107 for a 45-degree rhombus and 0.353553 for its half; these are areas in squared-module units, not centimetres or volumes of cavities ⁴. The record adds that doubling every length quadruples area, so scale and units are part of the mathematical answer ⁵.

Al-Khwarizmi says in his preface that the measurement of land is among the needs he wrote his book for ⁶. The record preserves no sentence in which he or al-Kashi explains why they treated solids or ceilings. For al-Kashi, Dold-Samplonius separates what the text says from what she infers: her explanation for his relatively careless treatment of simple muqarnas is an interpretation, not direct testimony about the author's intentions ⁷.

Many printed digits can suggest absolute accuracy, but the record says they do not promise absolute accuracy for every curved surface ⁸. For curved muqarnas the cell bases are summed and multiplied by a coefficient; the historian explains it as an average that unifies the calculation, not a magical constant valid for any chosen curvature ⁹.

Dold-Samplonius identifies a projection inconsistency: equating the roof area with the area on which the muqarnas is built conflicts with inclined roofs in the definition and the examples ¹⁰. She also does not know how some of the values given for the intermediate elements were obtained, and her reconstructions yield different numbers ⁹. The record says that reading should be neither a celebration that suppresses the inconsistency nor a wholesale dismissal because it exists ¹⁰.

Dold-Samplonius finds that one step of al-Kashi's method conflicts with the inclined roofs in his own definition and examples. Take a position: should a student still learn the method? What is the strongest objection to your position, and how do you answer it?

• Mensuration (ilm al-misaha): the rule for the cone and pyramid and the truncated case, as the record reports them.
• Al-Kashi's modular muqarnas mensuration: the Figure 8 values, the average coefficient, and what the historian does not know.
• The Key to Arithmetic: the book that carries the method, and what the museum has read of it.
• Al-Khwarizmi: the record of the author of the mensuration chapter.

Read Mensuration (ilm al-misaha) and Al-Kashi's modular muqarnas mensuration. Make a paper pyramid and compare its volume with the one-third rule, and write what doubling a length does to an area.