SOLUTION: We use π to calculate area of circles. The ancient Egyptians had a method, documented around 1650 BC in the Rhind Mathematical Papyrus, of calculating the area of a circle wit

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Question 1025322: We use π to calculate area of circles. The ancient Egyptians had a method, documented around 1650 BC in the Rhind Mathematical Papyrus, of calculating the area of a circle without π as we now know it. They took a square with side length equal to the diameter of the circle (Figure 1), trisected each side and removed the corner triangles (Figure 2) and used the remaining octagonal shape (Figure 3) to approximate the area of the circle. Using this method, what would be the calculated area of a circle with diameter 9 units?

https://www.mathcounts.org/sites/default/files/u5328/Egyptian%20Circle.jpg

Pi is the ratio of a circle’s circumference to its diameter. Using the octagonal shape from the ancient Egyptians’ calculations, calculate the ratio of the perimeter of the shape to the diameter of the circle it represents. What is the absolute difference between this value and π? Express your answer as decimals to the nearest hundredth.
Answer by ikleyn(52781) About Me  (Show Source):
You can put this solution on YOUR website!
.
1.  The area of the remaining octagonal shape is

    9%5E2 - 4%2A%28%283%2A3%29%2F2%29 = 81 - 2*9 = 63 sq.units against pi%2A4.5%5E2 = 63.585 sq. inits.



2.  The perimeter of the remaining octagonal shape is

    4%2A3+%2B+4%2A%283%2Asqrt%282%29%29 = 12+%2B+12%2Asqrt%282%29 = 28.971 units.


    The ratio of the perimeter of the remaining octagonal shape to the diameter of the circle is 

    28.971%2F9 = 3.219  (approximately).

    The difference between this value and pi is 0.077 (approximately).