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This article is designed to help with the calculation of distance and angles.

Real life applicationReal life application

While on a hiking trip, a certain hiker comes up to an impassible object or area.

The current bearing is 200 degrees. The hiker changes course by 20 degrees (compass set to 180 degrees) for 30 minutes. To get around what ever was blocking the path.

For the sake of discussion, the person walks on average 1 mile every 45 minutes - depending on backpack load and terrain. At 30 minutes the person is about 3/4 (.75) mile from the change of direction.

How far off course is the hiker?

__Solution:__Use the change of direction as the center of a circle, meaning multiple the distance traveled by 2. So the diameter of the circle is 1.5 miles, with the hiker on the outer edge.

The change of angle was by 20 degrees. So use the original path as 0 degrees.

Multiple the diameter of the circle by 3.14. To make this easy, just use 3.

1.5 X 3.14 = 4.71 circumference of the circle

1.5 X 3 = 4.5 circumference of the circle

Using 3 instead of 3.14 is going to be less accurate, but its something the person can do in their head while on the move.

This is where it might get a little tricky if the hiker is trying to do the math in their head.

Divide 4.5 by 360 - because a circle has 360 degrees in it. To make this easy, use 36, then add a 0 to the front of the number.

4.5 / 360 = .0125

4.5 / 36 = .125

The .0125 is the correct number. If 36 is used, be sure to add a 0 to the front.

Multiple 20 (for 20 degrees) by .0125 (for each degree) = .25 miles.

The hiker is about 1/4 mile off course.

__Example 2:__This time lets remove all the details and just show the math.

Hiker changes course by 33 degrees for 2 miles.

__Using 3 instead of 3.14__

2 X 2 = 4 (circle 4 miles in diameter with change of direction as the center)

4 X 3 = 12

12 / 36 = .333

Add 0 to the front of .33 = .0333

.033 X 33 = 1.089 miles off course

Because 3 was used instead of 3.14, that distance is going to be a little off.

__Using 3.14 instead of 3__

2 X 2 = 4 (circle 4 miles in diameter with change of direction as the center)

4 X 3.14 = 12.56

12 / 360 = .0348

.033 X 33 = 1.151 miles off course

To be as accurate as possible, 3.14 should always be used instead of 3.

To be as accurate as possible, 3.14 should always be used instead of 3.

The solution now gives the hiker 2 sides to a right triangle. With one side being 2 (distance traveled), and the other side being 1.151 (distance from hiker to original path).

On every right triangle, all three sides added together equal 180 degrees, with one side being 90 degrees.

We know that the hiker changed course by 33 degrees, so subtract 33 from 90. The hiker needs to change their compass by 57 degrees, hike 1.151 miles and they will be back on their original path.

__However:__Unlike a real triangle, in this example the short side has curve in it, which is equal to the radius of the circle used.

**You try it:**

Hiker changes direction by 25 degrees and hikes 2 miles.

Multiple 2 X 2 to give you the diameter of a circle.

4 * 3.14 = 12.56 miles in circumference

12.56 / 360 (or 36 and add your 0 back) = .034 miles per degree around the circle

.034 X 25 = .85 miles off course.

90 - 25 = 65 degrees

Change the compass by 65 degrees, hike .85 miles and you will be back on course.

__But then again.__Who says the hiker needs to go back to their exact course? Lets say that the hiker wants to continue on to base camp. Now we have 2 sides of a right triangle to calculate the distance from the hiker to the base camp in a straight line.