How to find the angle of clock hands - Geometry

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°.

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).

At four o’clock the minute hand is on the 12 and the hour hand is on the 4. The angle formed is 4/12 of the total number of degrees in a circle, 360.

4/12 * 360 = 120 degrees

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.

There are degrees in one complete revolution of a circle. There are minutes in one hour.

Create a fraction out of these two quantities to use later as a conversion rate:

Between the and there are minutes, so multiply this by the conversion rate to solve for the number of degrees:

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 .

You can think of a clock in two ways:

1. Out of 12 hours, or

2. In terms of a circle with

If you try to solve it in terms of #1:

Goal: Find the angle measurement between the hour and the minute hands. We only want to find the degrees between the hours of 9 and 12

So we are looking at 3 hours out of the 12 total hours on a clock.

As a fraction:

So that means that the clock hands are making an angle that is 1/4 of the clock (which is a circle). So knowing that a circle has in it,

1/4 of a circle is .

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2. If you think of the clock as a circle first you can determine the angle that the clock hands create very quickly.

Since there are in a circle, every hour that passes is a movement of . So knowing that, the clock will be moving 3 hours:

Like any circle, a clock contains a total of . Because the clock face is divided into equal parts, we can find the number of degrees between each number by doing . At 5:00 the hour hand will be at 5 and the minute hand will be at 12. Using what we just figured out, we can see that there is an angle of between the two hands. We are looking for the larger angle, however, so we must now do .

By dividing the clock into pieces, we can determine that the angle between the two hands is .

Within a clock, just like any circle, there are 360 total degrees.

Within an clock, there are 60 total minutes.

Each minute that passes, the minute hand advances 6 degrees. The hour hand advances .5 degrees. But since it is 9am on the dot, we will just be using the minute hand to count the degrees.

Since the minute hand is covering a total of 15 minutes in between it and the hour hand, we can do the math:

=

On first glance, this problem may seem simple in that the angle between the 3 and 6 on a clock is one quarter of a circle or 90 degrees. However, you must take into account that in the half hour that has passed from 3:00 to 3:30, the hour hand has moved half the distance between the 3 and 4. To find the degrees, simply divide the total number of degrees in a circle by 12 to find the degrees between each consecutive number, and then multiply that number by 2.5 because you have half the distance to the 4, and the the full distance to the 5 and the full distance to the 6. Thus,

To solve this question recall that a clock is a cicle and a circle is comprised of 360 degress.

To find how many degrees lie between each of the twelve numbers on the clock, divide 360 by 12.

.

From here, we will multiply 30 degrees by 4 to find the degree angle of for the hands of the clock at 4:00.

Another approach is to simply divide 4 by 12 to determine what percentage of the circle the hands will cover, and then multiply that number by 360, which is the full degrees in a circle. Thus,

.

It is important to recall that a clock is a circle and a circle is comprised of 360 degrees. Therefore, to calculate how many degrees are between the 12 and the 3 we need to set up a ratio.

The degree measure between each number is,

.

Therefore, to find the degree measure for three numbers would be,

.

If the hour hand is on the 12 and the minute hand is on the 3 it would create a right angle which is, .

It is important to recall that a clock is a circle and a circle is comprised of 360 degrees. Therefore, to calculate how many degrees are between the 12 and the 3 we need to set up a ratio.

The degree measure between each number is,

.

Therefore, to find the degree measure between 6 numbers it would be,

.

If the hour hand is on the 12 and the minute hand is on the 6 it would create a straight line which is, .

With one hand at o'clock and the other at o'clock, the configuration encompases of the complete of a full circle.

Therefore, we can determine the angle of the configuration as such:

There are 360 degrees in a circle. If you divide that by four then you get 90 degrees and if you divide 60 minutes by 4 you get 15 minutes. Therefore every 15 minutes on a clock represents 90 degrees. If I go clockwise from 12 there are 45 minutes or three lots of 15 minutes. If 1 lot of 15 minutes equals 90 degrees then,

The minute hand will be on the number 9, which would form a 90-degree angle with an hour hand pointing right at 12.

The hour hand has moved three-quarters, or 75% of the way from 12 to 1. Since the full 360 degrees of the circle represents 12 hours, the segment representing the hour between 12 and 1 is

.

75% of that is

.

The total angle is the 90 degrees between 9 and 12, and the additional 22.5 degrees, or

We can automatically eliminate 1:00 and 1:15, since at those times the clock hands are less than 90 degrees apart. At 1:30 the hour hand is halfway between the 1 and the 2. Since there are 30 degrees between each number on the clock face, the hour hand is 30 + 15 = 45 degrees away from 12. The minute hand is on 6, which is 180 degrees from 12. The two hands form a 180 - 45 = 135 degree angle.

At 2:45, the minute hand is on the 9, which is 90 degrees away from 12 to the left. The hour hand is three-quarters of the way between 2 and 3. Since there are 30 degrees between each number, this hand is degrees away from 12 to the right. The angle formed is 90 + 82.5 = 172.5 degrees.

At 3:30, the minute hand is squarely on the 6, and the hour hand is halfway between the three and the four, as shown below:

As seen in the diagram, the degree measure from one hour mark to the next is one twelfth of , or . The degree measure from the six to halfway between the three and four is - this is the angle the two hands form.

The angle measure of one full revolution around the clock face is , since the clock face is circular. Looking at the numbers which indicate the time, one full revolution around the clock face also corresponds to . To find the angle measure between the minute hand and hour hand, consider the minute hand as the initial side of the angle and the hour side as the terminal side. The hour hand has rotated to the position, so we can deduce the angle measure of the minute and hour hands by setting up and solving a proportion:

Hence, the angle measure between the minute hand and hour hand of a clock set at is .