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. Help him by completing the following table:
$\begin{array}{cccccc}No.ofspokes& 4& 6& 8& 10& 12\\ Anglebetweenapairofconsecutivespokes& {90}^{\circ }& {60}^{\circ }& \dots & \dots & \dots \end{array}$


(i) Are the number of spokes and the angles formed between the pairs of consecutive spokes in inverse proportion?
(ii) Calculate the angle between a pair of consecutive spokes on a wheel with 15 spokes.
(iii) How many spokes would be needed, if the angle between a pair of consecutive spokes is ${40}^{\circ }$ ?

Answer 1

Here the number of spokes are increasing and the angle between a pair of consecutive spokes is decreasing. So, it is a inverse proportion and angle at the centre of a circle is ${360}^{\circ }$ .
When the number of spokes is 8,
then angle between a pair of consecutive spokes = $\frac{{360}^{\circ }}{8}={45}^{\circ }$
When the number of spokes is 10,
then angle between a pair of consecutive spokes = $\frac{{360}^{\circ }}{10}={36}^{\circ }$
When the number of spokes is 12,
then angle between a pair of consecutive spokes = $\frac{{360}^{\circ }}{12}={30}^{\circ }$
$\begin{array}{cccccc}No.ofspokes& 4& 6& 8& 10& 12\\ Anglebetweenapairofconsecutivespokes& {90}^{\circ }& {60}^{\circ }& {45}^{\circ }& {36}^{\circ }& {30}^{\circ }\end{array}$

(i) Yes, the number of spokes and the angles formed between a pair of consecutive spokes is in inverse proportion.
(ii) When the number of spokes is 15, then angle between a pair of consecutive spokes =$\frac{{360}^{\circ }}{15}={24}^{\circ }$
(iii) The number of spokes would be needed = $\frac{{360}^{\circ }}{{40}^{\circ }}=9$