Question

Rehman is making a wheel using spokes. He wants to fix equal spokes in such a way that the angles between any pair of consecutive spokes are equal.

Number of spokes	$4$	$6$	$8$	$10$	$12$
Angle between a pair of consecutive spokes	${90}^{\circ }$	${60}^{\circ }$

How many spokes would be needed, if the angle between a pair of consecutive spokes is 40°?

Answer 1

Here the number of spokes are increasing and the angle between a pair of consecutive spokes is decreasing.

So, number of spokes and the angle between a pair of consecutive spokes is in inverse proportion.

We know that the angle formed at the centre of a circle is $360°$.
When the number of spokes is $4$, then angle between a pair of consecutive spokes is $90°$
For $6$ spokes, the angle between a pair of consecutive spokes is $60°$

For $8$ spokes, let the angle between a pair of consecutive spokes be $x$:
$\Rightarrow 6\times 60°=8\times x$
$\Rightarrow x=\frac{6\times 60°}{8}=45°$

Similarly, for $10$ spokes:
$\Rightarrow 6\times 60°=10\times y$
$\Rightarrow y=\frac{6\times 60°}{10}=36°$

For $12$ spokes:
$\Rightarrow 6\times 60°=12\times z$
$\Rightarrow z=\frac{6\times 60°}{12}=30°$

Number of spokes	$4$	$6$	$8$	$10$	$12$
Angle between a pair of consecutive spokes	${90}^{\circ }$	${60}^{\circ }$	${45}^{\circ }$	${36}^{\circ }$	${30}^{\circ }$

When the number of spokes is $4$, then angle between a pair of consecutive spokes is $90°$

Let for $p$ spokes, the angle between a pair of consecutive spokes be $40°$

$\Rightarrow 4\times 90°=p\times 40°$

$\Rightarrow p=\frac{4\times 90°}{40°}=9$

Hence, $9$ spokes will be needed.