Pi in Ancient Egypt

Geometry Level 2

While ancient Egyptians never had a generalized idea of π , \pi , they had this procedure for estimating the area of a circle: Take 8 9 ths \frac{8}{9}^\text{ths} of the diameter and then square it.

What is the implicit value of π \pi used in this procedure?

256 81 \frac{256}{81} 81 64 \frac{81}{64} 64 81 \frac{64}{81} 22 7 \frac{22}{7}

Jason Dyer by Jason Dyer

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1 solution

Jason Dyer Staff
Oct 21, 2016

Given a circle of radius r , r , 8 9 ths \frac{8}{9}^\text{ths} of the diameter squared is ( 8 9 2 r ) 2 = 256 81 r 2 . \left(\frac{8}{9}2r\right)^2 = \frac{256}{81} r^2.

Since the area of a circle is π r 2 , \pi r^2 , and the Egyptian estimate uses 256 81 r 2 , \frac{256}{81} r^2, the Egyptian value of π \pi is implicitly 256 81 . \frac{256}{81} .

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