Draft: this lesson was written with AI help and has not yet been read by a human editor.
1 What your class asks
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
Curriculum expectations, with their official codes
- E2.3 solve problems involving the perimeter, circumference, area, volume, and surface area of composite two-dimensional shapes and three-dimensional objects, using appropriate formulas
2 Before you start
A thought to carry into the lesson.
3 The lesson
How do you measure a shape built from simple shapes?
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.
Go deeper
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.
Why they did it
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.
Often misunderstood
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.
Think about it
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?
Into the museum
- 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.
Try this
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.
4 Carry it further
Where this lesson leads in the museum.
5 Check yourself
Nothing here is marked or sent anywhere. It is for you.
In your own words: what does Mensuration (ilm al-misaha) have to do with this lesson?
Compare with the card
A chapter on measuring. Mensuration is the science of measuring areas and volumes, and al-Khwarizmi gave it a chapter of his book. The record lists his rules: the area of a triangle is its height times half its base, and the circumference of a circle is the diameter times three and a seventh, which he calls "a convention among people, not a necessity". Which of these rules can you check by measuring a real circle?
In your own words: what does Al-Kashi's high-precision circumference computation have to do with this lesson?
Compare with the card
Sixteen decimal places for the circle. MacTutor dates al-Kashi's Treatise on the Circumference to 1424 and describes him expressing twice the ratio of circumference to diameter in sexagesimal and decimal forms with high precision. Okten reports the decimal value 6.2831853071795865, an approximation to twice the ratio, not to pi itself. The record calls this an improvement of an inherited calculation, not the discovery of pi. Which circle has a circumference equal to that number?
I can now
Notes
- 1. Ontario Ministry of Education, The Ontario Curriculum: Mathematics
- 2. al-Khwarizmi, Kitab al-Jabr wal-Muqabala, ed. Mosharafa and Ahmad (Cairo, 1939), p.78
- 3. al-Khwarizmi, Kitab al-Jabr wal-Muqabala, ed. Mosharafa and Ahmad (Cairo, 1939), p.79
- 4. al-Khwarizmi, Kitab al-Jabr wal-Muqabala, ed. Mosharafa and Ahmad (Cairo, 1939), p.81
- 5. al-Khwarizmi, Kitab al-Jabr wal-Muqabala, ed. Mosharafa and Ahmad (Cairo, 1939), p.82
- 6. Dold-Samplonius: measuring muqarnas by al-Kashi, pp193-and196
- 7. Al-Kashi — MacTutor, Key
- 8. Dold-Samplonius: measuring muqarnas by al-Kashi, pp209-213
- 9. Kashi — Sadettin Okten, mathematics