I’m back with the final installment of ‘Where The *** Am I?’, Calculating a Grid Square.

I’m assuming that you’ve read and understand the information in Part 1 and Part 2b.

For this example, I’ll continue to use the same coordinates as in the previous segments.

**Latitude: 41.8820670**

**Longitude: -87.6278160**

As with the previous examples in parts 1 and 2b, these coordinates should come back to the intersection of State and Madison in the heart of Chicago’s Loop.

There are six digits in a typical Grid Square location. To be accurate, typical Grid Squares can be four or six digits in length. Actually, a “grid square” can be comprised of the first two digits alone, but you might as well say somewhere really big on planet earth. At best, here in ‘the vicinity’ of the United States, a two digit grid square might narrow your location down to somewhere in North America.

The grid resolution you choose to use depends on how detailed you wish to be regarding your location. It is possible to actually drill down as deep as an 8 digit Grid Square, even 10 theoretically. That, however, is overkill for the vast majority of Amateur Radio operations.

**Getting Started**

**The First Digit.
**

For our first digit, we simply take our Decimal Longitude and add 180 to it, then divided by 20. If one is in the western hemisphere, longitude will be a negative number (i.e. preceded with a minus (-) sign.

-87.627816 plus 180 equals 92.32184 ( -87.627816 + 180 = 92.32184 )

Now, divide the result, 92.32184 by 20:

92.32184 divided by 20 equals 4.6186092 ( 92.3218400/20 = 4.6186092 )

The number to the left of the decimal point, **4**, is the integer value of 4.6186092 derived in the previous step. That integer, 4, is a place-holder value for our first Grid Square digit. More on that in a bit.

OK great, but we’re not done yet! We need to derive another value to be used later when we calculate our third Grid Square digit.

To obtain that value we must perform Modulo Division on 4.6186092, the value we obtained in the previous step. Remember, however, the goal is to do this calculation manually.

So, to perform modulo division manually, we start by dropping the number to the left of the decimal point, 4, which we’ve already identified as the integer of 4.6186092. This leaves us with 0.6186092, the leftover digits after the initial divide by 20 in the previous step.

Since we are not using a scientific calculator or spreadsheet with a Modulo function in this exercise we, again, do it manually.

Multiply 0.6186092, by the same divisor used to obtain 4.6186092. We had divided 92.32184 by 20, so now we’ll multiply 0.6186092 by that same divisor, 20.

0.6186092 times 20 equals 12.327184 ( 0.6186092 * 20 = 12.327184 )

File that number, * 12.327184*, away for use later on in the process.

**The Second Digit.**

This process is identical to the first, except that we will be using our Decimal Latitude, adding 90 to that value, and then dividing that number by 10.

41.882067 plus 90 equals 131.882067 ( 41.882067 + 90 = 131.882067 )

Now divide that result, 131.882067, by 10:

131.882067 divided by 10 equals 13.1882067 ( 131.882067/10 = 13.1882067 )

As with the first digit, the number to the left of the decimal point, **13**, is the place-holder value for our Second Grid Square digit.

As with the first digit calculation, we’re not done yet, We again need another value to be used later to calculating our fourth Grid Square digit.

Just as we did in the first digit calculation, we must perform Modulo Division on 13.1882067, the value we obtained in the previous step.

So, we start by dropping the number to the left of the decimal point, 13, which we’ve already identified as the integer of 13.1882067. This leaves us with 0.1882067, the leftover digits after the initial divide by 10 in the previous step.

We now multiply 0.1882067 by the same divisor used to obtain 13.1882067. We divided 131.8820670 by 10, so now we’ll multiply 0.1882067 by that same divisor, 10.

0.1882067 times 10 equals 1.882067 ( 0.1882067 * 10 = 1.882067 )

File that number, * 1.882067*, away for use later on in the process.

**The Third Digit.
**

**This is where we start to use the secondary results derived during the calculations for the first two digits to obtain our third and fourth Grid Square digits**

Start by taking the results of the Modulo Division (which we did manually) derived while calculating our first digit, ** 12.327184. **In this process, we will start by dividing this value by 2.

12.327184 divided by 2 equals 6.186092 ( 12.327184 /2 = 6.186092 )

As in previous calculations, we need the integer of 6.186092, which is the same as simply taking the number to the left of the decimal point, **6. **Not a place-holder, this is the actual value for third Grid Square digit.

However, as with the preceding processes, we’re not done yet. We again have to do some Modulo Division to later obtain, in this case, the fifth digit of our Grid Square.

Take the digits to the right of the decimal point, 0.186092, we use Modulo Division by multiplying this value by the original divisor in this section, 2. This gives us a value to save for use later, * 0.372184*, as we calculate our fifth Grid Square Digit.

**The Forth Digit.**

Start by taking the results of the Modulo Division (which we did manually) derived while calculating our second Grid Square Digit, *1.882067.*

In this process, there is really no need to go through the process of division, because anything divided by one is the same number. We do require the integer of * 1.882067*, which is, 1. As with the third digit, it’s not a place-holder, this is the actual value for our Forth Grid Square digit,

**1**.

However, as with the preceding processes, we’re not done yet. We again have to do some Modulo Division to later obtain, in this case, the sixth digit of our Grid Square.

Take the digits to the right of the decimal point, 0.186092, we use Modulo Division by multiplying this value by the original divisor in this section, 1. This gives us our value to save for later, * 0.372184*.

Now before we continue, it is important to note that we have all but completed 4 digit grid square which is a common/general locator. To complete that, we need to map our place-holder values to alpha characters. We do this by correlating the numerical values to letters of the alphabet or, as I jokingly refer to it, our ‘Little Orphan Annie Secret Decoder Ring’. Don’t worry if you don’t get the reference…

0 (zero) correlates to the letter “A”, 1 (one) to the letter “B”, 3 (three) to the letter “C”, and so on until we end up at 25 (twenty-five), which correlates to the letter “Z”

With that in mind**, **we convert digits one and two, 4 and 13, which become E and N. Remember that digits three and four, 6 and 1, were not ‘place-holder’, but the actual digit.

Putting all four together in order, we get a Grid Square of **EN61**.

This four digit grid square represents a geographical area of 20 degrees of longitude by 10 degrees of latitude. This particular grid, EN61, is depicted in the image below, taken from the website QTH Locator.

As can be seen in the image, this is a fairly large geographical area but is, nonetheless, a common reference in amateur radio communications. When you are talking about HF (High Frequency), a.k.a. *‘short wave’* propagation, this is enough in most cases to see how far your signal is traveling or, how far away another station is from your location.

We can, however, do better. A four digit grid square can be further divided into twenty-four, 5 minute (**0.083333** degrees) of longitude and Ten, 15 second (**0.00416665** degrees) of latitude.

**The Fifth Digit.**

Start by using the result of the Modulo Division (which we did manually) while calculating our Third Grid Square Digit, *0.372184.*

We divide this value by 0.08333, representing five minutes of longitude.

0.372184 divided by 0.083333 equals** 4.66225865
**( 0.372184/0.083333 =

*)*

**4.66225865**Taking the integer of * 4.66225865*, gives us,

**4**, which is a ‘place-holder’ value for our fourth grid square digit. If we run that past our previously discussed “

*decoder ring”*, we get the letter,

**E**.

Note: *It is simply a coincidence of this example, that our first and forth grid square digits are the same.*

**The Sixth Digit.**

Here we do the same as we did with the fifth digit, taking the result of our modulo division, except in this case, from the Forth Grid Square Digit calculations, ** 0.882067**.

We divide this value by **0.0416665**, representing 15 degrees of latitude.

0.882067 divided by 0.0416665 equals 21.16969268

( 0.882067/0.0416665 = 21.16869268 )

Taking the integer of * 21.16969268*, gives us,

**21**, which is a ‘place-holder’ value for our forth grid square digit. If we run that past our previously discussed “

*decoder ring”*, we get the letter,

**V**.

Putting it all together, we now have a 6 digit Grid Square: **EN61EV.**

Once again using the website QTHLocator, it can be easily seen that the new grid square covers a geographical area that is substantially smaller.

Generally speaking, a 6 digit Grid Square is about as tight of resolution as anyone typically goes.

That said, in researching this topic deeper, I discovered a paper written by Les Peters, N1SV, on calculating a forth pair, further reducing down the geographical area under the Maidenhead system.

Les explains the fourth pair this way:

** ‘Each field can be further **subdivided

**character**

*into (10) 30 seconds (0.008333 degrees) longitudinal by (10) 15 seconds (0.004166) latitudinal zones. The first*

*encodes the longitude and the second encodes the latitude with numbers “0” through “9”.’***Source**: *Les Peters, N1SF…How to calculate your 8-digit grid square*

**The Seventh Digit.**

To begin, we have to return to the fifth digit calculations. We need to process the leftover part of * 4.466225865*, the integer of which, 4, was our fifth digit. Specifically, we need to process that number 0.466225865 by multiplying it by

*the original divisor from that same ste, (modulo division) gives us*

**0.008333,****0.38852**

0.038852 divided by 0.008333 equals **4.662426497**

( 0.038852/0.008333 = **4.662426497** )

Taking the integer of 4.662426497, we obtain our seventh Grid Square Digit, **4**.

**The Eighth Digit.**

The process used in the seventh digit is repeated except that our leftover via modulo division is 0.007075, taken from the sixth digit calculations, and our divisor is 0.004166.

0.007075 divided by 0.004166 equals **1.697191551
**( 0.007075/0.004166 =

**1.697191551**)

Taking the integer of 1.697191551, we obtain our eighth Grid Square Digit, **1**.

Putting it all together now results in an eight digit grid square of. **EN61EV41**.

To see the results of these extra digits, we must use a different mapping web site, k7fry.com.

Plugging in the eight digit grid square, we can see the results in the following image.

As can be seen, the total geographical area is once again reduced. This time to a few square blocks. Looking closely at the image, we can see that our original coordinates, which resolve to the intersection on State and Madison, located in Chicago’s ‘Loop’, are in fact located within this grid square.

**Beyond the scope of this posting…**

Astounding to me, I discovered through the use of the K7FRY website, a ten digit grid square is possible. Unfortunately, I have been unable to find any clear documentation on the process, other than it might be another iteration such as that used in calculating the seventh and eight digits.

However, the K7FRY website also featured a click on a location, and so it was possible to learn ones ninth and tenth grid square digits. The results were interesting as can be seen in the following image.

The geographical area has gone from several square blocks, to perhaps something less than a couple of hundred square feet. This would, in many cases, permit one person to actually see another, even in a crowded urban setting such as the Chicago Loop.

To be sure, I don’t know anyone who uses an eight, let alone a ten, digit grid square. It is, nonetheless, interesting to know it is possible and, how to obtain them with a little basic math. Well, eight digits at least.

That’s it! If I have screwed up somewhere with the math, please let me know.

Until next topic…