Is the shape of a triangle determined by its perimeter and area? In other words, if two triangles have the same area and the same perimeter, are the triangles similar? [1]
It’s plausible. A triangle has three degrees of freedom: the lengths of the three sides. Specifying the area and perimeter removes two degrees of freedom. Allowing the triangles to be similar rather than congruent accounts for a third degree of freedom.
Here’s another plausibility argument. Heron’s formula computes the area of a triangle from the lengths of the sides.
Here s is the semi-perimeter, half of the sum of the lengths of the sides. So if the perimeter and area are known, we have a third order equation for the sides:
If the right-hand side were 0, then we could solve for the lengths of the sides. But the right-hand side is not zero. Is it still possible that the sides are uniquely determined, up to rearranging how we label the sides?
It turns out the answer is no [2], and yet it is not simple to construct counterexamples. If all the sides of a triangle are rational numbers, it is possible to find a non-congruent triangle with the same perimeter and area, but the process of finding this triangle is a bit complicated.
One example is the triangles with sides (20, 21, 29) and (17, 25, 28). Both have perimeter 70 and area 210. But the former is a right triangle and the latter is not.
Where did our algebraic argument go wrong? How can a cubic equation have two sets of solutions? But we don’t have a cubic equation in one variable; we have an equation in three variables that is the product of three linear terms.
What third piece of information would specify a triangle uniquely? If you knew the perimeter, area, and the length of one side, then the triangle is determined. What if you specified the center of the triangle? There are many ways to define a center of a triangle; would some, along with perimeter and area, uniquely determine a triangle while others would not?
Related posts
- Six triangle properties in one equation
- Perimeter, radii, and sides
- How you define center matters a lot
[1] Two triangles are similar if you can transform one into the other by scaling and/or rotation.
[2] Mordechai Ben-Ari. Mathematical Surprises. Springer, 2022. The author sites this blog post as his source.