Objectives:
Writing algebraic statements from word problems. Estimating π.
Background:
Although the irrationality of π was not known in the ancient world, various civilizations had their
own estimations for the value of π in order to answer practical questions using mathematics.
Currently students use 3.14 to estimate the value of π or reference their calculator. In this lesson,
student will use problems and solutions found in Ancient Egyptian papyrus to determine what
value the Ancient Egyptians used to estimate π.
Although the two exercises presented here are relatively straightforward, students often have a
difficult time writing expressions from word problems. This activity also provides students with
an ability to appreciate ancient societies’ development of and sophisticated use of mathematics.
They could also note how “boring” problems plagued students in math classes for thousands of
years!
Course:
High School Geometry
Standards:
CCSS.Math.Content.7.G.B.4
Know the formulas for the area and circumference of a circle and use them to solve problems;
give an informal derivation of the relationship between the circumference and area of a circle.
CCSS.Math.Content.HSA.CED.A.4
Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving
equations. For example, rearrange Ohm's law V = IR to highlight resistance R.
CCSS.Math.Content.HSG.GMD.A.1
Give an informal argument for the formulas for the circumference of a circle, area of a circle,
volume of a cylinder, pyramid, and cone. Use dissection arguments, Cavalieri's principle, and
informal limit arguments.
Review or Prerequisite knowledge:
 Multiple estimations for π
 Formula for area of a circle
 Formula for area of a square
 Writing algebraic expressions from word problems
Activities:
Activity 1:
The following question appeared as problem #50 in the Ahmes Papyrus:
A circular field has a diameter 9 khet1. What is its area?
1
A khet is a measure of length equal to approximately 52.5 meters or 57.4 yards.
The solution is also included:
Take away one-ninth of the diameter, namely 1: the remainder is 8. Multiply 8 times 8; it
makes 64. Therefore it contains 64 setat of land.
Each of these statements can be translated into algebraic expressions and equations. Students
should do so and relate their equations to their own understanding of how to calculate the area of
a circle given the diameter. From there, an estimation of π according to Ancient Egyptians
(256/81) can be calculated.
Solution:
1 2
8 2
𝐴 = (𝑑 − 𝑑) = ( 𝑑)
9
9
𝑑2
8 2
𝜋⋅
= ( 𝑑)
4
9
𝜋 64
=
4 81
𝜋=
256
∼ 3.16
81
Notice how the assumed estimate for π is 3.16 and not 3.14 as we use. Also note how strikingly
accurate the value is.
Activity 2:
But where did the Ancient Egyptians determine this formula? This activity comes from problem
48 in the Ahmes Papyrus and derives a different formula for area of a circle. The papyrus shows
this problem where the diameter of the circle is nine. However, we will use the spirit of the
problem do derive the area of the circle using a general diameter 𝑑.
In this problem, a square is circumscribed about a circle. Four isosceles triangles are drawn at the
corners of the square so that the vertices of the base of the triangle trisect the sides of the square.
These triangles are removed. The remaining figure is an octagon circumscribed about the circle.
d/3
d/3
d/3
d/3
From here, the area of the circle can be estimated by the area of the newly formed irregular
octagon. The area of the octagon is calculated by adding the squares and triangles that form it.
d/3
d/3
d/3
d/3
𝑑 2
1 𝑑 𝑑
𝑑 2 7 2
𝐴 = 5 ( ) + 4 ( ⋅ ( ) ( )) = 7 ( ) = 𝑑
3
2 3 3
3
9
Question for group discussion.
64
What is the relationship between these two formulas? The first formula for the area is 81 𝑑 2 . The
7
63
second is 9 𝑑2 . If we rewrite the second, we see it’s formula is 81 𝑑 2 so both formulas are very
close. In fact, in problem 48, the Ahmes papyrus sets 𝑑 = 9 and determines the area of the
octagon to be 63 square units which is close to 64 or 82 .
In this example, we can calculate π as well.
7 2 1 2
𝑑 = 𝜋𝑑
9
4
𝜋=
28
∼ 3.11
9
This value is not as close as the other estimate, but nevertheless we see that the Ancient
Egyptians did not have a regular estimate for π nor did they recognize or use irrational numbers.
References
Beckmann, P. (1971). A history of pi. New York: Barnes and Noble Books.
Howard, C.A. (2009/2010). Mathematics problems from Ancient Egyptian papyri. Mathematics
Teacher, 103(5), 332-339.