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I was wondering if there is a formula that could generate the values of the sides of a triangle where his area equals to his perimeter. I only found that if the triangle is equilateral then $$l=\frac{12}{√3}$$ where $l$ is the side of the triangle.

Thanks for support

Peterix

P.S. There is a similar problem here

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marked as duplicate by Blue, Daniel Fischer, Adam Hughes, Behaviour, daw Aug 11 at 20:18

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2 Answers 2

Let $S$ be the area of a triancle. Let $P$ be the perimeter of a triancle. Let $r$ be the inscribed circle radius of a triangle.

Since $S=\frac12 r P$ we have $r=2.$ It gives us information for certain types of triangles. for exemples. For a right triangle we have $$r=\frac{a+b-c}{2}=2$$ Since $$c^2=a^2+b^2$$ we have $$a+b=2+\frac{ab}{4}$$

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Can you please tell me what are S,P and r? –  Peterix Jan 25 at 21:52
    
@Peterix I edited my answer –  nadia-liza Jan 25 at 22:00
    
Thanks a lot for help! –  Peterix Jan 25 at 22:04

Given any triangle $T$ there is a triangle $T'$ similar to $T$ such that the area of $T'$ is the same as the perimeter of $T'$.

Suppose the perimeter of $T$ is $P$ and the area of $T$ is $A$. Then dilate $T$ by a factor of $\frac{P}{A}$ to produce the triangle $T'$. (That is, multiply all the lengths by $\frac{P}{A}$.)

two triangles

The perimeter of $T'$ will be the perimeter $T$ multiplied by $\frac{P}{A}$ to give: $$ \text{Perimeter of } T' = \frac{P}{A}\cdot P = \frac{P^2}{A} $$ The ara of $T'$ will be the area of $T$ multiplied by $(\frac{P}{A})^2$ to give: $$ \text{Area of } T' = \left(\frac{P}{A}\right)^2 A = \frac{P^2}{A} $$

For example, an equilateral triangle with all edges equal to 1 has area $\sqrt3/4$ and perimeter 3. The equilateral triangle with perimeter equal to its area is dilated by $3 \div (\sqrt3/4) = 12/ \sqrt3$, so its side lengths are all $12/\sqrt3$.

Another example: a 3-4-5 triangle has perimeter 12 and area 6. The similar triangle with the area equal to its perimeter is dilated by $12/6 = 2$. Thus the new triangle is a 6-8-10 triangle and has area and perimeter 24.

A final example: a right-angled triangle with short sides $a$ and $b$ has perimeter $P = a+ b+ \sqrt{a^2+b^2}$ and area $A = \frac12 ab$. Dilate this by $P/A$ to give sides of: $$ \begin{align} a \cdot \frac{a + b + \sqrt{a^2 + b^2}}{\frac12 ab} &= 2\left( \frac{a}{b} + 1 + \sqrt{\left(\frac{a}{b}\right)^2 + 1}\right)\\ \text{and } \quad b \cdot \frac{a + b + \sqrt{a^2 + b^2}}{\frac12 ab} &= 2\left( \frac{b}{a} + 1 + \sqrt{\left(\frac{b}{a}\right)^2 + 1}\right)\\ \text{and }\quad\sqrt{a^2+b^2} \cdot \frac{a + b + \sqrt{a^2 + b^2}}{\frac12 ab} &= 2\left(\sqrt{\left(\frac{a}{b}\right)^2 + 1} + \sqrt{\left(\frac{b}{a}\right)^2 + 1} + \frac{a}{b} + \frac{b}{a}\right) \end{align} $$

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