A analog clock is divided up into 12 sectors, based on the numbers 1–12. One sector represents 30 degrees (360/12 = 30). If the hour hand is directly on the 10, and the minute hand is on the 2, that means there are 4 sectors of 30 degrees between then, thus they are 120 degrees apart (30 * 4 = 120).

Because it's an 8 hour clock, each section of the clock has an angle of 45 degrees due to the fact that .

When the clock reads 1:30 the hour hand is halfway in between the 1 and the 2, and the minute hand is on the 4 (at the bottom of the clock). Therefore, between the hour hand and the "2" on the clock there are  degrees and between the 2 and the 4 there are  degrees. Finally, 

If creating a picture helps, draw a circle and place 8 at the top where 12 normally is. Then put 2, 4, and 6 at the positions of 3, 6, and 9 on a normal 12-hour analog clock. 1, 3, 5, and 7 go halfway in between each even number. 

Now, because each section is equally spaced, and because there are 8 sections we simply divide the total number of degrees in a circle () by the number of sections (8). Thus:

When the clock reads 8:15 the minute hand is on the 3 and the hour hand is just past the 8.

Each section of the clock is 

 degrees.

From 3 to 8 then there are 150 degrees. However, the hour hand has moved a quarter of the way between the 8 and the 9, or a quarter of 30 degrees.  and so 

Each section of the clock is , and by  the hour hand has gone three quarters of the way between the  and the . Thus there are  between the hour hand and the  numeral. The minute hand is on the , and there are  between the  and the . So in total there are  between the hands

The number of degrees between each numeral on the clock face is equal to the number of degrees in a circle divided by the number of sections:

At  the  hand has gone  way through the  between the and the . Thus there are only  left between it and the 3. There are 120 degrees between the 3 and the 5, where the minute hand is, so the total amount of degrees between the hands is:

Remember there are  in each hour long section of the clockface

.

When the clock reads 6:30 the minute hand is on the 6, and the hour hand is halfway between the 6 and 7.

Thus the number of degrees between the hands is 

When the clock reads  on this clock, the hour hand will be  of the way between the  and the . Since there are , evenly sized, sections of this clock each section has:

. And . At  the minute hand will be one-quarter of the way around the entire dial. 
Thus the hands are  from each other

The entire clock measures 360°. As the clock is divided into 12 sections, the distance between each number is equivalent to 30° (360/12). The distance between the 2 and the 3 on the clock is 30°.  One has to account, however, for the 10 minutes that have passed. 10 minutes is 1/6 of an hour so the hour hand has also moved 1/6 of the distance between the 3 and the 4, which adds 5° (1/6 of 30°). The total measure of the angle, therefore, is 35°.

To find angular distance between the minute and hour hand, first find the position of each. Using  as a reference (both hands straight up), we can calculate the difference in degree more easily.

First, remember that a circle contains , and therefore for the minute hand, each minute past the hour takes up  of angular distance.

For the hour hand, each hour in a 12-hour cycle takes up  of angular distance, which means each minute takes up of distance for the hour hand.

Thus, our distance from (our reference angle) in degrees for the minute hand can be expressed as , where is the number of minutes that have passed.

Likewise, our distance from in degrees for the hour hand can be expressed as , where is again the number of minutes that have passed.

Then, all we need to do is find the positive difference between these two measurements, , and we have our angle.

This looks more complicated than it is! Fortunately, once our equation is set up, the problem is as easy as plug and play. First, let's find the measure of the minor arc between the minute hand and the reference. We can ignore hours (as each hour returns the minute hand to our reference angle) and just look at minutes.

Now, find the same distance, but for the hour hand. Here, we must consider both hours and minutes by using a fraction:

Lastly, we find the difference between these two references (remembering that our answer should be positive):

Thus, the hands are apart at .