Question

Please help me find out how to obtain the measure of the following angles in a clock when the hour is a) 8:54, b) 5:11 and c) 8:03 (the angles that are formed by the hour hand and the minute hand.

Answer 1

Answer:

A) 84

B) 84

C) 222

Explanation:

The angle between the hour hand and the minute hand is shown below.

Now the angle Θ is equal to the angle subtended by the hour clock with respect to 12 o'clock and the angle between 12 o'clock angle the minute hand.

Θ = | the minute hand - the hour hand |

A.

The angle between 12 o'clock and 8: 00 is

This angle is

$\frac{8\colon\text{ 00}}{12\colon00}\cdot360$

in other words, 8 / 12 of a full circle.

The above simplifies to give

$\frac{8\colon\text{ 00}}{12\colon00}\cdot360=240^o$

Now we find the angle between 12 o'clock and the minute hand.

This angle is

$\frac{54}{60}\cdot360$

in other words, 54/60 of a full circle.

The above simplifies to give

$\frac{54}{60}\cdot360=324^o$

Hence, the angle between the minute hand and the hour hand is

$324-240=84^o$

The angle is 84 degrees.

B)

The angle we are looking for is shown in the sketch below.

The angle between 12 o'clock and 5: 00 is

$\frac{5\colon\text{ 00}}{12\colon00}\cdot360=150^o$

Now we find the angle between 12 o'clock and the minute hand.

The angle between 12 o'clock and 11 minutes is

$\frac{11}{60}\cdot360=66^o$

Therefore, the angle between the hands is

$150^o-66^o=84^o$

The angle is 84 degrees.

C)

The angle we are looking for is shown in the sketch below.

The angle between 12 o'clock and 8:00 is

$\frac{8}{12}\cdot360=240^o$

The angle between 12 o'clock and 3 minutes is

$\frac{3}{60}\cdot360=18^o$

Therefore, the angle between the hands is

$240^o-18^o=222^o$

The angle is 222 degrees.

Hence, to summerise

A) 84

B) 84

C) 222