Convert angles from the sexagecimal system to centesimal one

Question

How do you convert angles from the sexagecimal system to centesimal one?

For example

63 degrees 14 minutes 51 seconds reduced to centesimal ??

Here's how it's done but I don't understand the steps: image at postimg

Answer 1

First you convert the measurement to purely degrees. There are 60 seconds per minute and 60 minutes per degree.

So $51$ seconds is equal to $\frac{51 \textrm{sec}}{60 \textrm{sec/min}} = 0.85 \textrm{min}$ . Your angle can now be written as: 63 degrees, 14.85 minutes.

Converting 14.85 minutes to degrees, we again divide by 60:

14.85 minutes is $\frac{14.85 \textrm{min}}{60 \textrm{min/deg}} = .2475 \textrm{deg}$. Your angle is now written as: 63.22475 degrees.

Convert one more time by the appropriate constant.

Answer 2

I write the conversion factors so that it looks clearer:

$$51'' = 51'' \cdot \frac{1'}{60''} = \left(\frac{51}{60}\right)' = \left(\frac{17}{20}\right)' = 0.85'$$

So,

$$14'51'' = 14.85' = 14.85' \cdot \frac{1^\circ}{60'} = 0.2475^\circ$$

and thus,

$$63^\circ 14'51'' = 63.2475^\circ = 63.2475^\circ \cdot \frac{1 \text{ rt. angle}}{90^\circ} = 0.70275 \text{ rt. angle} $$ $$= 0.70275 \text{ rt. angle} \cdot \frac{100^g}{1 \text{ rt. angle}}= 70.275^g$$