I will know the length of two sides of any triangle that I use, but I will not know any of the angles. I know how to find the length of the third side if I knew the angle where I am sitting, but how can I quickly find the included angle where I am sitting with basic geometric tools or something else?

  • $\begingroup$ Are you asking how to figure out the unknown side without knowing the included angle? $\endgroup$ Nov 30, 2014 at 6:24
  • $\begingroup$ It's more like how can I find the included angle in the field. $\endgroup$
    – Parable
    Nov 30, 2014 at 6:26
  • $\begingroup$ Shoover's answer is correct. You cannot figure out the third side without more information. Is it possible that you meant the triangle to be a right triangle, as it appears in your drawing? $\endgroup$
    – Kaj Hansen
    Nov 30, 2014 at 6:30
  • $\begingroup$ No I didn't mean it as a right triangle and I will probably never be able to figure out if there is a right angle in the triangles I use so it's best to assume that it's not a right triangle. $\endgroup$
    – Parable
    Nov 30, 2014 at 6:41
  • $\begingroup$ To figure out the angle "quickly in the field with basic tools", I will suggest either of two instruments. One is a magnetic pocket bearing compass, but a more basic model might also work. You would also need a calculator and/or trig tables. You may be able to measure the angle within 2 degrees of arc. For the compass to be useful there must be visible targets for your sights, and the angle must be in a nearly horizontal plane. A more accurate measurement could be made with a sextant, but that might not be a "basic tool", more in the "something else" category. $\endgroup$
    – LouisB
    Nov 30, 2014 at 7:22

5 Answers 5


Basic Answer:

You can't.

Here's what you can know

Given only the length of two sides of a triangle, the length of the third side is not fixed. Let a and b represent the lengths of the two known sides such that $a \geq b$. Let c represent the length of the unknown side, the length of c must fall within

$a - b < c < a + b$

Based on the example triangle you give, the third side, c, must be

\begin{align} 475 - 390 <& c < 475 + 390 \\ 85 <& c < 865 \end{align}

More Detail

With the two given lengths, we can construct a segment and a circle. It is irrelevant where we choose to position the segment. It's endpoints can lie anywhere with the caveat that it has a length exactly equal to the first known side. For your example let the segment have length 475.


With the second side we can represent all possible endpoints as the points of a circle. This makes sense since a circle is the set of all points a given distance away from the center point. So the circle would have radius 390.

Segment + Circle

Well, you want to know the third side of the triangle, but the third side -- without any other information about the triangle -- could be any segment which starts at the free endpoint of our original segment and has its other endpoint on the circle. You can see why this means that there is more than one possible segment length because not all such segments have equal length. Here are a few examples.

ex. 1 ex. 2 ex. 3

Closing Remarks

If you want to calculate the third side of the triangle, you need more information than simply two sides. For example, if you know the triangle is a right triangle, or if you know the measure of the included angle between the two known segments, then you can determine the length of the third side.

  • $\begingroup$ Thank you for your response. Again this does not answer my question. If you look at the second sentence in my post you will know what I'm trying to figure out. $\endgroup$
    – Parable
    Nov 30, 2014 at 14:16
  • $\begingroup$ Your question is unclear. That is why people answering are having a difficult time giving you the information you want. This sentence: "I will know the length of two sides of any triangle that I use, but I will not know any of the angles. I know how to find the length of the third side if I know the angle where I am sitting, but how can I figure this out quickly "in the field" with basic tools or something else?" should be revised. $\endgroup$
    – Nick H
    Nov 30, 2014 at 15:41
  • $\begingroup$ @NickH - this isn't a math problem, it's a surveying issue. OP wants to know how to physically measure that angle. The distance is anecdotal. I mean, with no angle, you know, there's no solution. He wants ideas how to measure the angle in the field. I assume without the expensive equipment. +1 by the way. $\endgroup$ Nov 30, 2014 at 18:58

Knowing only the lengths of two sides of the triangle, and no angles, you cannot calculate the length of the third side; there are an infinite number of answers.

To illustrate, imagine that you have two fixed-length pieces of wood, and you drill a hole near the end of each one and put a nail through the hole. You can then rotate the pieces relative to each other, changing the angle between them, and thus changing the distance between their free ends.

  • 1
    $\begingroup$ @Parable \\ Hey, the man DID give you an answer: He told you that, without knowing the angle between the two lines which emanate from your position, you will not have enough information to solve your problem. "... take down your post ...[,]" indeed!? $\endgroup$ Nov 30, 2014 at 8:27

Do you have a compass? You could determine the bearing of each object. Find the difference. And then you have the included angle and can determine the distance.


You need more information to determine that.

If the angle between those measurements reach zero (this is, the extreme points line in a line in front of you), such unknown will be 475 - 390 = 85.

The opposite could happen too, 390 measured in one direction and 475 in the other direction. In that case that unknown would be 457 + 390 = 865.

You see, there is no formula to get you that unknown in the present setting. Any value between 85 and 865 would be reasonable depending on the shape of that triangle.

So my kind suggestion is to get hold on another piece of data. As it seems you are able to measure distances, the easiest thing to do is to move yourself to a new position (say 10 yards apart), measure the distances again to your reference points, and measure also the distance you moved. If this is possible for you, then a rephrasing of your problem would help those who want to help you here.

  • $\begingroup$ Thank you for your response. I most likely won't be able to move from my spot to get another reference point, but it's slightly plausible. I know this angle can be figured out with a theodolite. Buying one of these is impractical in the sense of money and weight. Is there another cheaper way to find the included angle where I am sitting? $\endgroup$
    – Parable
    Nov 30, 2014 at 14:28
  • $\begingroup$ Sorry, apart from moving I don't see another option. If you can move to another position, please ensure you get reasonable measurements; this is, they need to be reasonably different from the ones you have (or put other way, you have to move at least 40 or 50 yards from your original point if your measurements are only precise +- 1 yards, I believe). $\endgroup$
    – carlosayam
    Dec 1, 2014 at 3:21

$$x + 475 > 390$$ $$390 + 475 > x $$ $$390 + x > 475$$

so $x$ can be any number bigger than 85 or smaller than 865



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