The path traveled by the tip of the minute hand over the course of one hour is a circle of radius . The circumference of that circle is

.

The tip has traveled 10 inches since noon, so the fraction of the circle traveled is ,

and the number of minutes that have expired since noon is .

Therefore, to the nearest minute, the time is 12:16.

The clock face is divided into sixty equal parts, each minute. The minute hand is located on the 20 minute mark at 6:20, the hour hand located between the 30 minute mark and the 35 minute mark. When the minute hand goes all sixty minutes, the hour hand only moves five, so to figure out the location of the hour hand, we look at how much the hour has progressed, in this case 20 minutes, or one third of the hour. So the minute hand has moved one third of the way through the hour, so has the hour hand moved one third of the way through the five minutes, or, five thirds of a minute, which is one and two thirds minute, one minute forty seconds. That puts the hour hand at thirty minutes plus one minute and forty seconds—at 31min 40sec—which is 11min 40sec farther than the minute hand.

The distance between A's hands is simple to find, the minute hand is on the 6 at 30 minutes, halfway through the hour and the hour hand is resting just past the 3 at 15 minutes. What the minute hand travels in an hour, the hour hand travels in five minutes, so halfway through five minutes is  added to 15 minutes puts the hour hand at 17.5 minutes.

To find the distance we subtract 17.5 from 30:

Therefore A's hands have a 12.5 minutes distance.

The distance between B's hands is the same procedure, the minute hand is on the 7 at 35 minutes, 35/60 through the hour, and the hour hand is resting past the 4 at 20 minutes, so  is added to 20, which is approximately 23 minutes, To find the distance we subtract 23 from the 35:

putting B's hands 12 minutes apart.

The first thing you will have to do is determine the degrees that the hour hand is away from the minute hand. To help find the degrees of the hour and minute hand, you have to realize that when the minute hand is at 30 minutes, the hour hand is halfway between the hours of 10 and 11.  Using the fact that at 9:00, the hour and minute hands create a angle. Knowing that the hour hand is in between 10 and 11, which is HALFWAY between 9 and 12, the hour hand and the 12 create a angle. So adding that to the  the half circle creates from 12 to 6, then the total would be:

Using the degrees and the circumference of a circle, you can find the distance of the arc length with the following formula.

where,

The minute hand represents our radius therefore .

Therefore, our equation becomes:

A picture will help a bunch for this problem, so we begin by looking at a clock at 9:00

We should notice two things.  The first is that by drawing in the distance we wish to find, namely that between the ends of the two hands, we end up with a triangle.  Secondly, remembering that a full circle is 360 degrees, we see that the distance between the two hands spans 3 hours, or one fourth of the circle.  One fourth of 360 is 90, so the angle between our two hands is a right angle.

We now realize that we have a right triangle, and our desired distance is simply the hypotenuse, which we can name .  This means we can find our distance by utilizing the Pythagorean Theorem.  Plugging our known values in gives

Taking the square root gives our answer.

Our clock looks roughly like this:

Now, between every number on the clock, there are  or  degrees. Therefore, from  to , there are  degrees. To find the arc length, you use the equation:

Now, we know:

We know that . Therefore, we can write our equation:

Our clock looks roughly like this:

Now, between every number on the clock, there are  or  degrees. Therefore, from  to  there are  of these sectors. Therefore, there are  degrees. To find the arc length, you use the equation:

Now, we know:

We know that . Therefore, we can write our equation:

Our clock looks roughly like this:

Now, between every number on the clock, there are  or  degrees. We have a little trickier math to do, however. Let's subdivide the clock into  subsections instead. Each of these will have  degrees. Now, between  and , there are  such subsections. Since the hour hand is directly in the middle of  and , there is one more such subsection. Therefore, we have  total subsections or  degrees. To find the arc length, you use the equation:

Now, we know:

We know that .  Therefore, we can write our equation:

The angle measure between any two consecutive numbers on a clock is .

Call the "12" point on the clock the zero-degree point.

At 4:45, the minute hand is at the "9" - that is, at the  mark. The hour hand is three-fourths of the way from the "4" to the "5; that is,

Therefore, the angle between the hands is

, the desired measure.

At 4:40, the minute hand is on the 8, and the hour hand is two-thirds of the way from the 4 to the 5. That is, the hands are three and one-third number positions apart. Each number position is thirty degrees around the clock, so the hands form an angle of .