How Do You Calculate the Distance Between Adjacent Numbers on a Clock Face?

[chaoseverlasting]

Homework Statement

I'm trying to create a clock face in Python and can't quite figure out where I'm going wrong.

Statement: For a clock of radius 100 units, what will be the distance between two adjacent numbers?

If a turtle walks along the path joining two adjacent numbers, by what angle will it have to turn at every number.

Homework Equations

Trigonometric identities for tan / sin / cos.
Sum of angles in a polygon = 360 degrees

The Attempt at a Solution

An analog clock has numbers from 1 to 12, with an origin at the center.

Consider a figure where two adjacent numbers are joined by a straight line (say nos. 1 and 2)

Let the distance between the origin and the numbers 1 and 2 be 100 (the nos 1 and 2 are equidistant from the origin).

The angle made at the center is 360/12 = 30 degrees for a 12 sided regular polygon.

This means the triangle ABC (vertices at the origin, nos 1 and 2) is an isosceles triangle with angles 30, 75 and 75 degrees respectively.

Dropping a perpendicular from vertex B to point D on side AC results in an angle ABD of 60 degrees. Which implies that perpendicular BD is of length 100 * cos 60.

This then implies that side BC is of length $$\frac{100*cos60}{sin75}$$.The angle that the turtle will have to turn at every such vertex (1,2,3...12) is 30 degrees.

However, I think the length BC is incorrect.

Can someone please confirm?

 

[chaoseverlasting]

For a regular n sided polygon of side length a, and radius 'r', I am getting the following relationship:

$$a=2rsin\frac{\pi}{n}$$

This corresponds with my earlier derivation, however I'm still not generating the correct pattern.

Please help!

If the radius, r is 100, the side a should be approx. 51.77 units.

[Edit]
Again, to verify, the side length should be lesser than the length of the arc between these two points.
The length of the arc between them is approx. 52.36 units.

So, I'm guessing the side length seems to be alright.

Any thoughts on the angle to be turned, in that case?
[\Edit]

 

[Ray Vickson]

For a regular n sided polygon of side length a, and radius 'r', I am getting the following relationship:

$$a=2rsin\frac{\pi}{n}$$

This corresponds with my earlier derivation, however I'm still not generating the correct pattern.

Please help!

If the radius, r is 100, the side a should be approx. 51.77 units.

[Edit]
Again, to verify, the side length should be lesser than the length of the arc between these two points.
The length of the arc between them is approx. 52.36 units.

So, I'm guessing the side length seems to be alright.

Any thoughts on the angle to be turned, in that case?
[\Edit]

I can't see your problem: the arc-length between two numbers is $a = 52.35987758$, while the straight-line distance is $d = 51.76380902$; the angle between them is $\theta = 2 \pi/12 = 0.5235987758$ radians = $360/12 = 30$ degrees. Your formula for $d$ is correct; all you need to do is substitute in the numbers $r = 100, n = 12$, being careful to use radians as the angle measure when calculating the sine function in that specific formula (or else change the formula to handle degrees).