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There is a triangle whose area can be calculated just by knowing/measuring one of its sides.

If it is not an equilateral triangle, what it could be ?

Karri Chandrasekhar
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    Isosceles right triangle, perhaps? Or any other triangle where we know all the angles already. – Arthur Aug 23 '17 at 12:07
  • @Arthur, yes, why do not you go ahead and post your answer !:-) – Karri Chandrasekhar Aug 23 '17 at 12:18
  • @Arthur Only if we know which side is given. In the case of an isosceles right-triangle, we'd need to know whether the side given was the hypotenuse or a cathetus. – Tez LaCoyle Aug 23 '17 at 12:22
  • @T.Linnell, no it does not matter. If either one of them is given, we can calculate its area. Please check. – Karri Chandrasekhar Aug 23 '17 at 12:23
  • In that case, you'll be able to tell me what the area of an isosceles right triangle is if I tell you one of its sides has length $1$. – Tez LaCoyle Aug 23 '17 at 13:13
  • @T.Linnell, yes it would be 1/2 * 1 * 1 = 1/2 square units. – Karri Chandrasekhar Aug 23 '17 at 13:24
  • Wrong. The triangle I had in mind had a hypotenuse of $1$, so its legs have lengths $1/\sqrt{2}$ each and so the area is $1/2 * (1/\sqrt{2})^{2}$ = $1/4$ of a square unit. I never said the side I gave was one of the shorter sides. Do you see my point now? – Tez LaCoyle Aug 23 '17 at 13:29
  • @T.Linnell, yes, agreed, a subtle point missed out and perhaps I could have added it in the problem description itself, though while measuring a side, it is assumed/known that which side one measures. Thanks. – Karri Chandrasekhar Aug 23 '17 at 14:08

3 Answers3

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The area of a degenerate triangle can be calculated without knowing/measuring any of its sides.

The point is, it is not true that you can compute the area of any triangle by only measuring one side unless you actually have other measurements (i.e., additional information of some sort, such as knowing the triangle is equilateral, degenerate, etc.). If you ONLY have the measurement of one side, and no other information, you cannot determine the area.

MPW
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  • there is a special case, where it is possible. Please check Arthur's comment above ! – Karri Chandrasekhar Aug 23 '17 at 12:20
  • @KarriChandrasekhar : As I said, you then have additional information. Knowing that the triangle is isosceles, or is a right triangle, or knowing the angles, is indirectly telling you additional measurements. But I think I understand what you mean. – MPW Aug 23 '17 at 13:54
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Any triangle where we know all the angles will do, as long as we also know which side we're measuring.

The other two sides may then be found using the law of sines, and then the area can be found by any of a number of formulas using the side lengths and angles, such as Heron's formula, or $\frac12bc\sin A$.

Arthur
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By Heron's formula ($A = \sqrt{p(p-a)(p-b)(p-c)}$) it's necessary to set three independent conditions on side lengths in order to get a unique answer.

By specifying two of them, it's sufficient to know the measure of a single side to find the area of the triangle.

Blex
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  • but to know the value of p, we must have the measures of all three sides, which is not the case here ! – Karri Chandrasekhar Aug 23 '17 at 12:21
  • not necessarily, for example if $a = 2 c$, $b = \frac{2}{3} a$, just measure $c$ to also get $p$. – Blex Aug 23 '17 at 12:27
  • Hmmm..it seems the other relationships among the measures of the sides are assumed and hence you are concluding only one side's measurement is enough. Then it is becoming a puzzle :-), than a plain mathematical question, I think. – Karri Chandrasekhar Aug 23 '17 at 12:35