Measuring areas and surfaces

This lesson asks you to find the areas of trapezoids, rhombuses, kites and composite polygons by splitting them into shapes you know, and the surface areas of prisms and pyramids by adding the areas of their faces. The museum offers area rules from an early algebra book and a method for adding up the surfaces of a ceiling. ¹

Draw the shape on a grid and cut it into rectangles and triangles. Add the areas of the parts, then check by cutting it another way.

In the Key to Arithmetic, al-Kashi organised the measurement of muqarnas: he selects a module, counts elements by shape, and adds the surfaces of facets, roofs and connecting pieces ². In the definition the study discusses, a muqarnas is a staircase-like ceiling: a cell consists of two facets and a roof, and cells whose bases lie in one horizontal plane form a tier ³. Some visible pieces belong to cells and others fill the gaps between them, so counting the hollows in a photograph is not enough to determine their exposed surface area ³. For plane surfaces the facet bases are summed and multiplied by the height, the roof pieces are added, and then the tiers are combined ⁴.

The museum's records preserve no sentence in which al-Kashi says why he measured muqarnas. They show that his book places arches, vaults and buildings among the matters a calculator needs to understand ^{5 6}. In an earlier book, al-Khwarizmi says in his preface that the measurement of land is among the needs he wrote his book for ⁷.

Al-Kashi can be taken for the inventor of muqarnas, but builders already knew how to assemble the cells before his mathematical account, and the historian Dold-Samplonius explains that he did not need to teach his practical readers the whole craft from its beginning ⁸. The record describes his method as a calculation within an older architectural practice, not an attribution of muqarnas itself to one inventor ².

In class you split a composite shape two ways and got the same area. Al-Kashi's method sorts the pieces of a ceiling by shape and adds their areas. Why might a measurement of a ceiling go wrong if some pieces are counted as cells when they only fill the gaps between cells?

• Al-Kashi's modular muqarnas mensuration: the module, the classes of pieces, and how their areas are added.
• The Key to Arithmetic: the book that carries the method, and where its author placed it.
• Mensuration (ilm al-misaha): the rules for the triangle, the rhombus and the circle in al-Khwarizmi's chapter.
• Ghiyath al-Din al-Kashi: the record of the book's author.

Read the records on mensuration and on al-Kashi's muqarnas. Pick the rhombus rule, test it on a drawing, and then list the faces of a prism you can hold and add their areas.