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What Are Perfect Triangles?

In trigonometry, a perfect triangle is defined as a triangle that has a perimeter equal to its area. There are several ways of finding perfect triangles. A perfect triangle can be an equilateral, an isosceles, or a right triangle. This article will show you several examples!

Using the Pythagorean Triples to Find Perfect Triangles

The Pythagorean Theorem states that given a right triangle abc, the sum of the squares of the lengths of the sides is equivalent to the square of the hypotenuse.

The three positive integer values and whole numbers a, b, and c (representing the length of the hypotenuse of a right triangle and the two side lengths) together make a Pythagorean triangle or Pythagorean triple.

There is an infinite list of Pythagorean triples. If we list the first few Pythagorean triples, we can find a couple of examples of perfect triangles, where the perimeter is equal to its area.

(3, 4, 5)

(5, 12, 13)

(6, 8, 10)

(7, 24, 25)

Here we see that (6, 8, 10) is a perfect triangle: The perimeter (6+8+10) is equal to 24, and the area ((6 x 8) / 2) is also equal to 24.

However, the Pythagorean triple (3, 4, 5) is not a perfect triangle because the perimeter (3+4+5 = 12) is not equal to its area ((3 x 4) / 2 = 6).

(5, 12, 13) is another Pythagorean triangle whose perimeter is equivalent to its area. Can you find its area and perimeter?

Image of a perfect triangle using the Pythagorean triples

Image of a not perfect triangle using the Pythagorean triples

A Perfect Triangle With Rational Numbers

Pythagorean Triples are always formed by whole integer numbers. But, there are also examples of perfect triangles that use rational numbers that are not whole numbers. Take the special case of an isosceles triangle with lengths 7.8, 7.8, and 6.0. It is a perfect triangle because the perimeter, 21.6, is equivalent to its area, 21.6.

Image of a perfect triangle with rational numbers

Perfect Triangles Using Heron's Formula

Pythagoras wasn't the only famous Greek mathematician who developed a special theory of numbers. Hero of Alexandria was another Greek mathematician who came up with Heron's Formula, which gives you the area of a triangle from the length of its integer sides. In algebraic terms:

Heron's formula

If you are interested in advanced concepts, you can use this formula to solve for more perfect triangles!

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