Using a compass and triangulation to measure distance

In the wilderness, it is sometimes useful to be able to calculate the distance to a faraway object. Using a compass and basic trigonometry (don’t worry, we’ll show you how), you can easily estimate the distance to a faraway object.  This is possible using principles of geometry and triangulation – if we know any three of the sides or angles of a triangle, we can calculate the remaining angles and sides.  In the examples below, we’ll demonstrate two different methods to calculate the distance to an object.

Introduction

Using the illustration below, we will first (1) “walk off” two distances, (2) sight the faraway object in our compass from both points, and finally (3) record the angle for both points to use in our distance calculation.  Using the known distance (the number of steps we walked off) and the angles measured at each of the two points, there are a couple of easy ways to estimate the distance to the object.  We can use basic trigonometry (using a “tangent” table listed below) or we can draw the layout to scale on a piece of paper to get our distance.  The trigonometric method will produce the most accurate measurement but requires a table of tangents (or a calculator to calculate the tangent of the angles).  Using a scaled drawing on paper is not as accurate but requires nothing more than a compass and something to draw on (even the dirt on the ground would work for this purpose).

Using a compass and triangulation to measure distance

 

Triangulation without a map or drawing to scale

The triangulation method will use the following formula.  Don’t worry if it looks complicated – we’re going to break it down to make it very easy.  Our object is to find the distance from the observer’s point to the ship (marked “d?” in our illustration)

d = (Tan (90 – (A -B))) x Baseline

d = Distance to be calculated
Tan = Tangent value of the resultant angle (we’ll look this value up in a table)
A = Greater value of the two measured bearing angles (the greater of Angle A or Angle B)
B = Lower value of the two measured bearing angles (the lessor of Angle A or Angle B)
Baseline = Measured  distance between our two bearing angles (distance between Angle A and Angle B)

Step 1

Starting at the point marked “Observer” (in the illustration above), count off the steps to a point perpendicular to the faraway object (in our example drawing, we want to calculate the distance to the ship and will begin by counting the steps between the “Observer” position and the point marked “Angle A”).  You can place a stick in the ground at all points to aid in stepping off the distances.  You don’t have to walk precisely perpendicular to the object being measured but try to stay close to perpendicular and make sure you walk in a straight line.  We will call this first point “Point A” and the angle of that point “Angle A”.  The line that we walk on will be our “baseline”.  The longer the baseline (and hence, the more steps taken between the points), the more accurate the distance calculation will be.  The steps counted on the baseline will be used to calculate the “steps” to the object.  To calculate a distance to the object using feet, you can convert the measurement in steps to a feet typically by using an estimate of 3 feet per step (long step).

Step 2

From this first point (labelled Angle A), sight the object using your compass and note the degrees between your “baseline” (the line you walked down) and the line running towards the object.  This will give us the value of Angle A.  It may help to lock the compass, so it does not rotate, and point North (0 degrees) down your baseline path.

In the picture below,the red line is running parallel to our baseline path (the path we are stepping off) while the yellow line points towards the distant object.  The angle measured here is around 60 degrees (as marked by the red numerals on the compass dial).

Magnetic compass face showing degrees

Step 3

Return to your original “Observer” position and do the same thing but in the opposite direction.  Try to keep the distance (number of steps) close to the same as the distance you marked off in Step 1.  This second point will be “Point B” and the angle that you measure here will be “Angle B”.  It will help the calculation if you make this angle (or Angle A) as close to 45 degrees (a “right angle” – shaped like the letter “L”) as you can get.

Step  4

Add the steps to Point A and Point B together to get the total steps of our baseline (the distance between Point A and Point B).  In mathematical terms, this is the “known segment” of our triangle (the one side of the triangle that we know the length for) and is shown as “Baseline” in our formula.

Step 5

Note which of the two measured angles is greater.  This will become “A” in our formula.  The lessor of the two angles will be “B” in our formula.  Calculate (A – B) or in other words, subtract the lessor angle from the greater angle.  If Angle A was 45 and Angle B was 60, we would subtract 45 from 60 like this: 60 – 45 = 15.

Step 7

Now we will figure out the tangent value to look up in our Degrees to Tangent Ratio table.  To do this, simply subtract the value we calculated above (the difference between the two angles) from 90.  Continuing our example, our formula would be: 90 – 15 = 75.

Step 8

Look up the Tangent value in the table below.  In our example, we look up the value for 75 which according to our table, is 3.732.

Step 9

Finally, we calculate the distance to the faraway object by multiplying the baseline distance by our tangent value.  Continuing our example, let’s say the distance between Point A and Point B is 10 steps.  Multiply 10 by our tangent value (3.732 – the value we looked up in the tangent ratio table) to calculate the number of steps to the distant object which in our example would be: 10 x 3.732 = 37.32 steps.  The total distance to our faraway object is 37.32 steps.

For the math whizzes out there, our entire formula translates to this:

d = (Tan (90 – (A -B))) x Baseline

d = (3.732 (90 – (60-45))) x 10

Note that most calculators out there, including most calculator apps available on smartphones, include a Tan function.  Simply enter the difference between the two angles and press the Tan key on the calculator.  Then multiply the result by the baseline distance to get the distance to the target object.

Here’s the diagram again with all the values filled in.

Using a compass and triangulation to measure distance

Degrees to Tangent Ratio table

1 =  0.017
2 =  0.034
3 =  0.052
4 =  0.069
5 =  0.087
6 =  0.105
7 =  0.123
8 =  0.140
9 =  0.158
10 = 0.176
11 = 0.194
12 = 0.212
13 = 0.230
14 = 0.249
15 = 0.267
16 = 0.286
17 = 0.306
18 = 0.325
19 = 0.344
20 = 0.363
21 = 0.384
22 = 0.404
23 = 0.424
24 = 0.442
25 = 0.466
26 = 0.488
27 = 0.509
28 = 0.532
29 = 0.554
30 = 0.577
31 = 0.601
32 = 0.635
33 = 0.649
34 = 0.674
35 = 0.700
36 = 0.726
37 = 0.753
38 = 0.781
39 = 0.810
40 = 0.839
41 = 0.869
42 = 0.900
43 = 0.932
44 = 0.965
45 = 1.000
46 = 1.035
47 = 1.072
48 = 1.110
49 = 1.150
50 = 1.192
51 = 1.234
52 = 1.280
53 = 1.327
54 = 1.376
55 = 1.428
56 = 1.482
57 = 1.539
58 = 1.600
59 = 1.664
60 = 1.732
61 = 1.804
62 = 1.880
63 = 1.962
64 = 2.050
65 = 2.144
66 = 2.246
67 = 2.355
68 = 2.475
69 = 2.605
70 = 2.747
71 = 2.904
72 = 3.077
73 = 3.270
74 = 3.487
75 = 3.732
76 = 4.010
77 = 4.331
78 = 4.704
79 = 5.144
80 = 5.671
81 = 6.313
82 = 7.115
83 = 8.144
84 = 9.514
85 = 11.430
86 = 14.300
87 = 19.081
88 = 28.636
89 = 57.289
90 = Undefined

 

Triangulation using a simple drawing

Suppose you do not have a calculator nor the Degrees to Tangent ratio table readily available?  You can still calculate the distance to the faraway object using a scaled drawing of the triangle and rudimentary measurements of the distances calculated based on the drawing.  Many of the steps will be the same as in our example above but we’ll repeat those here for clarity.

Using a compass and triangulation to measure distance

 

Step 1

Walk off the distance between Point 1 and Point B (see diagram above) making sure you are walking closely to perpendicular to the object being sighted.  From both Point 1 and Point B, sight the object using your compass and note the degrees between your “baseline” (the line you walked down) and the line running towards the object.  This will give us the value of Angle A and Angle B.  It may help to lock the compass, so it does not rotate, and point North (0 degrees) down your baseline path.

Step 2

Draw your baseline on a piece of paper (or in the dirt on the ground).

Step 3

Draw lines matching the angles that you measured in Step 1.  Draw them out long enough so that they intersect (which is the point of faraway object we are measuring the distance to).

Step 4

Use a ruler (or some other object marked with equal increments) to measure the Baseline of your drawing.  Divide the stepped-off baseline measurement (number of steps between Point 1 and Point B) by your measurement on the paper.  This will give you the scaled distance variable.

Step 5

On the piece of paper, measure the distance between your observation point and the faraway object.  Multiple this measurement by the scaled distance variable to calculate the distance to the object.

The geometry behind the magic

For the math geeks, or those that are curious, here is the basic geometry behind the calculations.

Hypotenuse, adjacent, and opposite sides of a triangle

 

To define the trigonometric functions for the angle A (see diagram above), start with any right triangle that contains the angle A. The three sides of the triangle are named as follows:

  • The hypotenuse is the side opposite the right angle, in this case side h. The hypotenuse is always the longest side of a right-angled triangle.
  • The opposite side is the side opposite to the angle we are interested in (angle A), in this case side a.
  • The adjacent side is the side having both the angles of interest (angle A and right-angle C), in this case side b.

In ordinary Euclidean geometry, the inside angles of every triangle total 180°. Therefore, in a right-angled triangle, the two non-right angles (Angle A and Angle B in our example) total 90°.

Sine

The sine of an angle is the ratio of the length of the opposite side to the length of the hypotenuse. (The word comes from the Latin sinus for gulf or bay, since, given a unit circle, it is the side of the triangle on which the angle opens.) In our case

Sine (sin) calculation

Note that this ratio does not depend on the size of the particular right triangle chosen, as long as it contains the angle A, since all such triangles are similar.

Cosine

The cosine of an angle is the ratio of the length of the adjacent side to the length of the hypotenuse: so called because it is the sine of the complementary or co-angle.  In our case

Cosine (cos) calculation

Tangent

The tangent of an angle is the ratio of the length of the opposite side to the length of the adjacent side.  In our case

Tangent (tan) calculation

Why are sin, cos, and tan important?

 

Calculatiing sin, cos, and tan

 

The sin, cos, and tan are important because they allow you to work out angles when you know the lengths of the sides or to work out the lengths of the sides when you know the angles.

The acronyms “SOHCAHTOA” (“Soak-a-toe”, “Sock-a-toa”, “So-kah-toa”) and “OHSAHCOAT” are commonly used mnemonics to help remember these ratios.  The following illustration helps demonstrate how sin, cos, and tan are calculated.

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