Clock Math

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What is the measure, in degrees, of the acute angle formed by the hands of a 12-hour clock that reads exactly 3:10?

35°

55°

60°

65°

72°

Explanation

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°.

What is the measure, in degrees, of the acute angle formed by the hands of a 12-hour clock that reads exactly 3:10?

35°

55°

60°

65°

72°

Explanation

What is the measure, in degrees, of the acute angle formed by the hands of a 12-hour clock that reads exactly 3:10?

35°

55°

60°

65°

72°

Explanation

If a clock reads 8:15 PM, what angle do the hands make?

Explanation

A clock is a circle, and a circle always contains 360 degrees. Since there are 60 minutes on a clock, each minute mark is 6 degrees.

$\frac{360^o\ \text{total}}{60\ \text{minutes total}}=6\ \text{degrees per minute}$

The minute hand on the clock will point at 15 minutes, allowing us to calculate it's position on the circle.

$(15\ min)(6)=90^o$

Since there are 12 hours on the clock, each hour mark is 30 degrees.

$\frac{360^o\ \text{total}}{12\ \text{hours total}}=30\ \text{degrees per hour}$

We can calculate where the hour hand will be at 8:00.

$(8\ hr)(30)=240^o$

However, the hour hand will actually be between the 8 and the 9, since we are looking at 8:15 rather than an absolute hour mark. 15 minutes is equal to one-fourth of an hour. Use the same equation to find the additional position of the hour hand.

$240^o+(\frac{1}{4}\ hr)(30)$

We are looking for the angle between the two hands of the clock. The will be equal to the difference between the two angle measures.

$\angle=247.5^o-90^o=157.5^o$

If a clock reads 8:15 PM, what angle do the hands make?

Explanation

A clock is a circle, and a circle always contains 360 degrees. Since there are 60 minutes on a clock, each minute mark is 6 degrees.

$\frac{360^o\ \text{total}}{60\ \text{minutes total}}=6\ \text{degrees per minute}$

The minute hand on the clock will point at 15 minutes, allowing us to calculate it's position on the circle.

$(15\ min)(6)=90^o$

Since there are 12 hours on the clock, each hour mark is 30 degrees.

$\frac{360^o\ \text{total}}{12\ \text{hours total}}=30\ \text{degrees per hour}$

We can calculate where the hour hand will be at 8:00.

$(8\ hr)(30)=240^o$

$240^o+(\frac{1}{4}\ hr)(30)$

We are looking for the angle between the two hands of the clock. The will be equal to the difference between the two angle measures.

$\angle=247.5^o-90^o=157.5^o$

It is 4 o’clock. What is the measure of the angle formed between the hour hand and the minute hand?

$120^\circ$

$30^\circ$

$60^\circ$

$180^\circ$

$90^\circ$

Explanation

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

It is 4 o’clock. What is the measure of the angle formed between the hour hand and the minute hand?

$120^\circ$

$30^\circ$

$60^\circ$

$180^\circ$

$90^\circ$

Explanation

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

It is 4 o’clock. What is the measure of the angle formed between the hour hand and the minute hand?

$120^\circ$

$30^\circ$

$60^\circ$

$180^\circ$

$90^\circ$

Explanation

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 clock in the classroom reads 5:00pm. What is the angle that the hands are forming?

$150^{\circ}$

$180^{\circ}$

$50^{\circ}$

$360^{\circ}$

$50^{\circ}$

Explanation

Since the clock is a circle, you can determine that the total number of degrees inside the circle is 360. Since a clock has 12 numbers, we can divide 360 by 12 to see what the angle is between two numbers that are right next to each other. Thus, we can see that the angle between two numbers right next to each other is $30^{\circ}$ . However, the clock is reading 5:00, so there are five numbers we have to take in to account. Therefore, we multiply 30 by 5, which gives us $150^{\circ}$ as our answer.