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Decreasing the Circumference of Differently Sized Circles: A Counterintuitive Cricket Problem

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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.

Sumville's Village Cricket Pitch

Sumville's Village Cricket Pitch

The Village of Sumville and Its Cricket Pitch

The village of Sumville has a cricket pitch in the middle. The pitch is a perfect circle of radius 50 metres and is surrounded by a boundary rope going around the whole circumference. Unfortunately, overnight somebody steals 5 metres of this rope. As the rope no longer covers the entire circumference, the villagers decide to make their cricket pitch slightly smaller, reducing the circumference by 5 metres, hence also reducing the radius.

The Intergalactic Gods' Cricket Pitch

The Intergalactic Gods' Cricket Pitch

The Intergalactic Gods and Their Cricket Pitch

The Intergalactic Gods also have their own circular cricket pitch. As they are gods, their cricket pitch is much larger than Sumville’s and has a radius of 1,000,000,000 miles. They also have a boundary rope going around the circumference of their pitch and unfortunately have also had 5 metres of this rope stolen. Just like the villagers of Sumville, the Gods decide to make up for this by reducing the size of the circle so that the circumference is 5 metres less.

The Problem

Both cricket pitches have had the circumference of their pitches reduced by five metres, but which of the two pitches will end up reducing in radius the most?

Basic intuition would suggest that the villagers' pitch will reduce in radius the most. After all, they have lost 5 metres of circumference from a pitch which only has a radius of 50 metres. For the Gods, however, a 5 metre loss of circumference doesn’t seem like much when compared to a circle of radius 1,000,000,000 miles. Let’s do the maths to see if this is true.

The Math

For any circle, the circumference is calculated using the following formula:

Circumference = 2πr, where r is the radius of the circle.

So for our circles we have that:

Original circumference = 2π × original radius

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New circumference = 2π × new radius

As both pitches have lost 5 metres of circumference, the difference between the original circumference and the new one is 5, hence:

Original circumference – new circumference = 5

2π × original radius − 2π × new radius = 5

2π × (original radius – new radius) = 5

Original radius – new radius = 5 ÷ (2π) ≈ 79.6 cm

A Surprising Result

You can see from the formula that the difference in size between the original radius and the new radius is 79.6 cm, regardless of the starting size of the radius.

Therefore our intuition was wrong. For any circle of any size, if you remove 5 metres from the circumference, the radius will reduce by approximately 79.6 cm. This is true for the Smallville pitch, the Intergalactic Gods pitch and any size of circular pitch in-between.

A Counterintuitive Cricket Problem on the DoingMaths YouTube channel

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

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