The circumference and area of a circle

This lesson asks you to use the relationships among a circle's radius, diameter and circumference to explain the formula for the circumference, and to show the relationships among radius, diameter and area to explain the formula for the area. The museum offers one story: how al-Khwarizmi states the rules for circumference and area and how al-Kashi computed the circumference; constructing circles and real-life problems are for your class to bring. ¹

Draw a circle and mark its diameter and its half. What does the area rule in the record say about these quantities, and how does it help you explain the formula you know?

Al-Khwarizmi gave mensuration a chapter of his Book of al-Jabr and al-Muqabala ². In that chapter, as the museum's record reports it, the circumference of a circle is the diameter times three and a seventh ³. Its area is half the diameter times half the circumference, or the square of the diameter less a seventh and half a seventh of it ^{3 4}.

The museum's records preserve no sentence in which al-Kashi says why he computed the circumference. Iranica, as the record of his precise computation reports, connects his treatise with improving approximations used in astronomy and acknowledges older traditions ⁵. Ulugh Beg's scholarly environment offered support for astronomy and calculation ^{6 7}.

It is easy to imagine al-Kashi drawing a huge circle and measuring it. The record says that a small drawing cannot establish sixteen decimal places, and that his achievement was careful computation, not measuring a circle with a larger compass ⁸. MacTutor dates the work to 1424 and, in the record's words, this improved an inherited calculation and did not discover pi ⁸.

Al-Khwarizmi gives two rules for the area of a circle: half the diameter times half the circumference, or the square of the diameter less a seventh and half a seventh of it. Which rule would be easier to explain to a classmate who does not know the formula, and why?

• Mensuration (ilm al-misaha): all the rules of the chapter, including the circle's two area rules.
• Al-Kashi's high-precision circumference computation: what the record says about the computation and the limits of its precision.
• Al-Khwarizmi: the record of the author of the chapter.

Read Mensuration (ilm al-misaha) and then Al-Kashi's high-precision circumference computation. Take a circle of diameter seven units, compute its circumference and area with the two rules the first record gives, and compare them with what your own formulas give.