Converting Arc To Time and Time To Arc The Seafarer Way

Time and arc conversion has been around for many years, even before “push button” technologies existed. The relationship between the two holds immense significance in navigation, where sailors use them in their voyages.

But because of our modern equipment today, it is not apparent how the conversion from time to arc, and vice versa, is important. Many even do not know how to perform such calculations.

This post will help you understand the correlation between arc and time and the easy steps in converting arc to time and time to arc.

Principles for arc and time conversion

The fundamental principle of arc and time relationship is governed by the Earth’s shape and rotation rate. 

This concept is particularly relevant in celestial navigation, where heavenly bodies’ apparent movement across the sky is measured in arcs, and the duration to traverse that certain arc is measured in time.

Since the Earth rotates 360 degrees in 24 hours, we now have a reference formula for converting arc to time and time to arc. 

To simplify everything, take a look at the table below.

1 day	=	24 hours	=	360°
60 minutes	=	1 hour	=	15°
4 minutes	=	1°	=	60′
60 seconds	=	1′	=	15”

We will use this table for our conversion to understand the process better.

Tools you need for conversion

Since we’ll be doing old school, we only need two things for our calculations. Here are the following:

1. A piece of paper

2. A pen or pencil

3. Basic calculator, and/or

4. Nautical Almanac (Conversion of Arc to Time) – Free download on that link!

How to convert Arc to Time

Let’s say we have an arc of 329° 27′ 13” and want to convert it to time equivalent. Looking at the figure, we can say that its hour equivalent is closer to 24, the minutes near 30, and the seconds should probably close to 1.

If you want a quick solution for this, I created an arc-to-time converter that you can easily use. Otherwise, let’s proceed to manual calculations to learn the basics.

Using Basic Math

Here’s the formula for converting arc to time using basic math.

Degrees of arc / 15 (with the remaining decimals multiplied by 60) =  Hours

Minutes of arc / 15  (with the remaining decimals multiplied by 60) =  Minutes

Seconds of arc / 15 = Seconds

329° / 15 = 21.93333 (21 hours and 0.933 minutes) OR  21h 56m

27′  /  15 = 1.8 (1 minute and 0.8 seconds) OR 1m 48s

13” /  15 = 0.86 seconds

Add the answers and you’ll get 21 hours 57 minutes 48.86 seconds

Here’s a more detailed solution:

If you’re wondering why we’re multiplying the decimals by 60, that’s to convert them into minutes or seconds.

This is because one hour is 60 minutes, and one whole minute is 60 seconds.

Proportionally, half an hour, or 0.5 hours, is the same as 30 minutes, and a quarter of a minute or 0.75 minutes is the same as 45 seconds.

Using Tables from the Nautical Almanac

You won’t need a calculation for this one. Just follow the numbers to arrive at the final answer. We will use the same example above.

1. Find the arc degrees and note their corresponding value in time.

2. Check the minute degrees and note its corresponding value.

3. Find the seconds column and note its corresponding value.

4. Add them all together.

How to convert Time to Arc

Converting time to arc uses a different formula, not even reversed from the example above. But as long as you know the rules, you’re good to go.

If you want to do it automatically, here’s a time-to-arc converter that I created.

Let’s also try the basics using the formulas below to better understand the principles behind them.

Suppose we convert 17h 05m 39s to arc. Here’s how we do it.

Using Basic Math

Here’s the formula for converting time to arc using basic math.

Hours x 15 = hours

Minutes / 4 = minutes (with answers in degrees and minutes)

Seconds / 4 = seconds (with answers in minutes and seconds)

Here’s the solution:

In that solution, we divided the minutes and seconds by 4 to convert them into degrees. This is because 60 minutes (1 hour) equals 15°, and every degree equals 4 minutes of time.

So if we have 8 minutes of time, that is also equivalent to 2° of arc, and 30 minutes of time is the same as 7° 30’ of arc. 

Using Tables from the Nautical Almanac

We can also convert time to arc using the same Almanac table we used above. Using the same given, 17h 05m 39s, here are the steps to finding them.

1. To get the degrees, look for the 17h 05m time value or the CLOSEST figure. That would be 17h 04m. Take note of its corresponding value in degrees, which is 256°.

2. We now lack 01m, and to find that, head to the minutes column (0’-59’)and find the 1m value which is equal to 15’.

3. On the same column, look for 39s or the closest value less to it. In this example, that would be 0m 36s which equals 9s.

4. For the remaining 3 seconds, look for 3.00 seconds in the leftmost column. Here, we can see it’s equal to 45 seconds. 

5. Add them together, and you’ll get 256° 24’ 45’’.

Why should we learn arc to time and time to arc conversion?

1. Better understand celestial navigation – Calculating the ship’s position using the sun, stars, planets, and moon uses arc and time

2. Know a country’s time zone – Dividing a country’s longitude by 15 will give you its time based on UTC

3. Calculate sunrise and sunset – With the help of a nautical almanac, you can solve the sunrise or sunset given your longitude.

4. Backup knowledge in case electronic navigation fails – Electronics could crash anytime, and knowing the fundamentals provides critical backup.

5. Educational development – Part of your studies in school is arc and time conversion to solve various navigational problems.

6. Appreciate history – Imagine those sailors crossing the oceans and doing everything manually. Mad respect.

May the winds be in your favor.