Measuring land, circles and ceilings

This lesson asks you to solve problems involving perimeter, circumference, area, volume and surface area of composite two-dimensional shapes and three-dimensional objects, using appropriate formulas. The museum offers three cards on the measuring of areas, the circle and a muqarnas ceiling; the problems, the formulas and the calculations are for your class to do. ¹

Draw a composite shape, then divide it into shapes whose area formula you know. Work out each part, add them, and say which formula you used for each part and why. Check your answer by dividing the shape a different way.

In the chapter on mensuration, as the museum's record reports it, al-Khwarizmi sorts triangles into three kinds: right, acute and obtuse. For an acute triangle of unequal sides, of 13, 14 and 15 cubits, the foot of the height and the height itself are found by al-jabr and al-muqabala, giving a height of twelve and an area of eighty-four ^{2 3}. Another problem puts a square inside a triangle with sides of ten cubits and a base of twelve: the side of the square is made a "thing" and comes out at four and four-fifths cubits ^{4 5}.

The museum's records preserve no sentence in which al-Kashi says why he measured muqarnas. They show where he put it: the study locates muqarnas in the last section of chapter nine of Book IV of the Key to Arithmetic, on measuring structures and buildings, and that placement connects it with earlier plane-area problems ^{6 7}. Many roof shapes had already been discussed in Book IV's chapter two ⁸.

The long number credited to al-Kashi is not pi itself. Pi is the circumference divided by the diameter in ordinary plane geometry. Twice pi is the circumference of a unit-radius circle, and confusing them doubles the answer incorrectly. Okten reports al-Kashi's decimal value 6.2831853071795865, which is an approximation to twice the ratio, not pi itself ⁹.

A classmate says that 6.2831853071795865 is pi. What would you ask them to check, and why does the record's distinction between the ratio and twice the ratio change the answer to a problem about the circumference of a circle?

• Mensuration (ilm al-misaha): the three kinds of triangle, the square inside a triangle, and the other rules of the chapter.
• Al-Kashi's high-precision circumference computation: which number Okten reports, and what it is and is not.
• Al-Kashi's modular muqarnas mensuration: how the areas of a compound ceiling are added.

Read about mensuration, al-Kashi's circumference computation and al-Kashi's muqarnas mensuration. Choose one rule, apply it to a shape from your class, and compare your result with a formula you know.