Age
11 to 16
| Article by
NRICH team
| Published

Arclets explained



The Arclets problem set in September 2002 produced some very interesting and inspiring work from Madras College. This short article gives a flavour of the way that Sheila, Shona, Alison Colvin, Sarah, Kathryn and Gordan tackled the problem.

Each of the following shapes is made from arcs of a circle of radius r.

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Arclets Explained

What is the perimeter of a shape with $3$, $4$, $5$ and $n$ "nodes".

What happens when $n$ is very large?

Explain the relationship between your answers and the perimeter of the original circle

Here are arclets with $3$, $4$ and $5$ nodes:

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3-node solution

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Arclets Explained
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Arclets Explained

The angles at the centre of the inner circle are $60^{\circ}$

So the angles at the centre of the outer circles are $120^{\circ}$ and $240^{\circ}$ ($120^{\circ}+240^{\circ}= 360^{\circ}$.

We can therefore divide each circle into a $1/3$ ($120^{\circ}$ out of $360^{\circ}$) part and a $2/3$ ($240^{\circ}$ out of $360^{\circ}$) part.

The perimeter of the arclet is made up of 3 "inward" arcs of $1/3$ of the circumference and 3 "outward" arcs of $2/3$ of the circumference.

Perimeter=((3×13)+(3×23))πd=(1+2)πd=3πd=6πr

4-node solution

For the 4 node arclets the angles at the centre are $90^{\circ}$
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Arclets Explained

The inward arcs are $1/4$ of the circumference ($90^{\circ}$ - there are 4 inward arcs.

The outward arcs are $1/2$ of the circumference (4 lots of $45^{\circ}$ - there are 4 outward arcs.

Perimeter=((4×14)+(4×24))πd=(1+2)πd=3πd=6πr

5-node solution

Because the inner circle is surrounded by five outer circles there are 5 angles - all of $72^{\circ}$ at the centre.

Using the properties of isoseles triangles the outward arcs are $2/5$ of the circumference and the inward arcs are $1/5$ of the circumference.



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Arclets Explained
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Arclets Explained
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Arclets Explained
Perimeter=((5×15)+(5×25))πd=(1+2)πd=3πd=6πr

An image of part of the work Sarah, Kathryn and Gordon did to find the perimeter of the 5-node arclet is shown below

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Arclets Explained

6-node solution

Scanned diagrams showing the work of Sheila, Shona and Alison to find the perimeter of a 6-node arclet:

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Arclets Explained

 

Perimeter for 6 nodes=((6×16)+(6×26))πd=(1+2)πd=3πd=6πr

Finding the perimeter of the N - node arclet

We know that these are the equations for the perimeter of 3, 4, 5 and 6 node arclets:

Number of nodes

3 4 5 6


Perimeter


$ \quad 3\times 2 \pi r = 6 \pi r \quad $


$ \quad 3\times 2 \pi r = 6 \pi r \quad$ $ \quad 3\times 2 \pi r = 6 \pi r \quad $ $ \quad 3\times 2 \pi r = 6 \pi r \quad$

If we substitute $N$ for the node number we get (in every case):

Perimeter for N nodes=((N×1N)+(N×2N))πd=(1+2)πd=3πd=6πr


This is based on the fact that the angles at the centres of the circles will be $1/N$ of a full turn.

If $N$ is very large the node shape begins to look like a circle:

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Arclets Explained

In other words. No matter how many nodes the perimeter will always be 3 circumferences.