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How to Prove Pi Equals 2: A Mathematical Paradox

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I am a former maths teacher and owner of DoingMaths. I love writing about maths, its applications, and fun mathematical facts.

A unit semicircle.

A unit semicircle.

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.

Splitting Into Two Semicircles

Suppose 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.

Two semicircles of radius 1/2.

Splitting Into Four Semicircles

Let's split these up again so we now have four 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.

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.

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.

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.

This content is accurate and true to the best of the author’s knowledge and is not meant to substitute for formal and individualized advice from a qualified professional.

© 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.