# Angles And Their Measurements

I have an intuitive understanding of what degrees and radians are and what arc length, subtending, area of the sector and the derivation of the formula $$s = r \cdot \theta.$$ However due to lack of reasoning skills, I am unable to get started with these problems.

The problems go like this.

1. Find the angle between the minute hand of a clock and the hour hand when the time is (a) 5:25, and (b) 6:10.

2. Find the distance between the two points on the earth,when the arc joining these points subtends an angle $5$ minutes at the earth's centre, the radius of the earth being $6370$ kms.

3. The moon's distance from the earth is $360000$ kms and the diameter subtends an angle of $31$ minutes at the eye of the observer. Find the diameter of the moon.

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For the first two questions: you need to see the correspondence between the 60 minutes in an hour and 360 degrees in a whole circle... – J. M. Oct 23 '11 at 9:05
For the final pair of questions you need to see the correspondence between the $360\times 60$ minutes of arc and $2\pi$ radians in a whole circle – Henry Oct 23 '11 at 9:09
These are very good and sensible problems. You better acquire a thorough understanding of the issues occurring therein, so that problems of this kind become "no problem at all" before going on to more serious matters in mathematics. This has nothing to do with "reasoning skills" but with a good intuition for scales and linear relations between variables. – Christian Blatter Oct 23 '11 at 12:46

## 2 Answers

Hints:

1. the hour hand moves $2\pi$ radians in $12$ hours, while the minute hand moves $2\pi$ radians in $1$ hour.

2. $60$ minutes of arc is $1$ degree and $360$ degrees is $2\pi$ radians.

3. same as for 2.

Apply $s=r\theta$, remembering this is true when $\theta$ is in radians and $s$ is arc length.

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a) hour hand at $25\frac{25}{60}$ mark, minute hand at 25 mark, 60 marks to the circle so difference is $\frac{25}{60}\frac{1}{60}$

1. Find the circumference of the earth and multiply by the fraction of a circle which 5 minutes corresponds to.
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Would be glad if you spoon feed me with step by step solutions for both the problems!! – alok Oct 24 '11 at 12:49