I am a former maths teacher and owner of DoingMaths. I love writing about maths, its applications and fun mathematical facts.
A Unit Semicircle
Take a unit semicircle (a semicircle with a radius of one unit). This semicircle will have a base of length 2 and a curved edge half the size of a unit circle's hence equal to 1/2 × 2πr = 1/2 × 2π × 1 = π.
A Unit Semicircle
Splitting Into Two Semicircles
Suppose now instead we put two smaller semicircles on our base line. Each of these has base 1 and so r = 1/2. The curved length of each semicircle is therefore 1/2 × 2π × 1/2 = π/2. Put together, this gives us a total curved length of 2 × π/2 = π, the same as before.
Two Semicircles of Radius 1/2
Splitting Into Four Semicircles
Let's split these up again so we now have 4 semicircles each with radius = 1/4. Each semicircle has curved edge = 1/2 × 2π × 1/4 = π/4 and so together have a total curved length of 4 × π/4 = π again.
Four Semicircles of Radius 1/4
Splitting Our Line Further
In fact it doesn't matter how many semicircles we split our base into, the total curved length always equals π.
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If we have n semicircles, each semicircle has diameter 2/n, hence a radius of 1/n. Each semicircle therefore has a curved length of 1/2 × 2π × 1/n = π/n. The total curved length is then n × π/n = π.
Splitting to Infinity
The curious thing here is if we keep on splitting. Each split gets our semicircles looking more and more like a straight line which suggests that the limit as we approach infinity is a straight line. We've shown that the total length of the curved edges always equals π and we know that the straight line has length 2, so π = 2.
Many Semicircles on the Diameter
The Problem With This 'Proof'
But surely π = 3.141... not 2? Don't worry, π does indeed equal 3.141... . What we have done here is demonstrated how careful we need to be when dealing with limits, especially when we think about those limits pictorially.
The line of semicircles doesn't actually tend towards a straight line. It may look it at a certain scale, but as soon as we zoom in, we just get a row of semicircles again. It can be seen clearly from the first image (by comparing the diameter and the curved edge) that π > 2, hence our example is not really a paradox, just bad reasoning.
© 2021 David
Umesh Chandra Bhatt from Kharghar, Navi Mumbai, India on July 01, 2021:
Very interesting paradox. Actually while number of semicircles tend to infinite the relative difference of semicircle and their base diameter remains same and until that diminishes the limit does not work.