Calculating the distance to a point

Question

Hi!

I'm trying to find an equation to satisfy a problem I encountered.

Scenario:

I was estimating distances on points by eye. It struck me that if you know the distance to 1 point of interest that you could use degrees from a compass to calculate angles (or so I think) in order to find the distance to the unknown.

For example, I am x. Known point 1 is y. Unknown point is z.

If I know that the distance from x to y is 300 meters, and it is 45 degrees from me, and I know that z is 2 degrees from me, how can I satisfy the angles and find the distance to z from me?

I never took trig or geometry, but it seems to me that there must be a way to compute this.

Thanks for any help, as it has been boggling my mind!

Answer 1

While this may be difficult problem to interpret, I believe the equation that would be of most help to you is:

s = r * θ

Where r is radius of a circle (which it sounds like is 300 meters in your case), θ is the angle measured between your position and the position of interest (this will need to be converted from degree to radians by multiplying your degrees by (π/180)), and s is the arc length found by multiplying the two together.

Hope this helps a bit!

-Fred

Answer 2

Hi Justin,

Angle measures are usually identified by three points. So if you know the angle at X from Y to Z, which we'd label as angle YXZ or simply angle X, and the angle at Y from Z to X, which we'd call angle ZYX or simply angle Y, and you know the length from X to Y which we can call side XY or z, we can solve for all other angles and lengths.

The angles in a triangle in a plane sum to 180 degrees, so since X + Y + Z = 180 degrees, you also know angle Z equals 180 - X - Y.

The law of sines tells us that XZ / sin(Y) = XY / sin(Z), so a bit of rearranging gives us XZ = XY * sin(Y) / sin(Z) which equals XY * sin(Y) * sin(180 - X - Y). You were correct; you can calculate the distance from X to Z by measuring angles while standing at X and Y and measuring the distance between X and Y.