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Replacing Circles ala No Pi

Note: Sometimes old-but-not-olde math texts include some problems involving some interesting ideas. Pulled from a text designed for engineering students, last week's problems dealt with areas of land...this week's deal with circle properties.


Problem 1: Find the percentage error in taking the area of the square as equal to that of the circle?


Problem 2: Find the percentage error in taking 4PM as equal to the circumference of the circle. Assume the radius is 1. Would the final answer be affected by assuming a different value for the radius?

 

Source: H. Rice & R. Knight's Technical Mathematics With Calculus, 1966


Hint: Problem 1 is quite straightforward...do you know the areas of the two shapes?

Problem 2 is more difficult...how to find the length of segment PM? Note: the special triangles available.

 


Solution Commentary: Problem 1 leads to an interesting extension: The current radius of the circle is 4...what should it be to make the two areas equal?

Problem 2 may require the use of some trig laws...at least I could find no other way (i.e. pure geometrical relationships). If you can solve this problem without trig, please share your solution with me (and overcome my frustration).