Explain Heron’s Formula

Heron’s formula (also known as Heron’s formula) gives the area of a triangle when the lengths of all three sides are known in geometry. It is named after the Hero of Alexandria. Unlike other triangle area formulas, no angles or other distances in the triangle must be calculated first. According to the Heron’s formula, the area of any respective triangle whose sides have lengths a, b, and c is where, s is the semi-perimeter that is obtained of the triangle; that is,

History of Heron’s Formula

Heron (or Hero) of Alexandria is credited with the formula, and proof can be found in his book Metrica, published around 60 CE. It’s been proposed that Archimedes knew the formula over two centuries before, and because Metrica is a compilation of the ancient world’s mathematical information, it’s conceivable that the formula predates the reference given in that work.

Proof of Heron’s Formula

Cyclic quadrilaterals were used in Heron’s original proof. Other claims make use of trigonometry, as seen below, or the triangle’s incenter and one excircle, or De Gua’s theorem (for the particular case of acute triangles).

Trigonometric proof using the law of cosines

A modern proof, that states the use of algebra and is quite different from the one provided by Heron which is added in his book Metrica. Let a, b, c be the sides of the respective triangle, and α, β, γ are the angles of the opposite of those sides. When applying the law of cosines we get the final formula.

Algebraic proof using the Pythagorean theorem

Any triangle with altitude or height h cutting base c into d + (c − d). The following proof is very related to one given by the great Raifaizen. By Pythagoras theorem we can state that b2 = h2 + d2 and a2 = h2 + (c − d)2. Subtracting the same gives us a2 − b2 = c2 − 2cd. The above equation allows us to express d in terms of the sides of the same triangle.

Special Case in Heron’s Formula

For the field of a cyclic quadrilateral, Heron’s formula is a special case of Brahmagupta’s formula. Heron’s and Brahmagupta’s formulas for the region of a quadrilateral are both special cases of Bretschneider’s formula. By setting one of the quadrilateral’s sides to zero, Heron’s formula can be obtained from Brahmagupta’s or Bretschneider’s formulas. Heron’s formula is a variant of the formula for calculating the area of a trapezoid or trapezium using only its sides. Setting the smaller parallel side to zero yields Heron’s formula. Using a Cayley–Menger determinant to express Heron’s formula in terms of the squares of the distances between the three given vertices,

Fun Fact

Hero (or Heron) of Alexandria, a Greek Engineer, and Mathematician from 10 to 70 AD, is credited with devising the formula. He created the Aeolipile, the first known steam engine, among other items, but it was regarded as a toy!

“S” symbolization in Heron’s Formula

The s in Heron’s formula stands for the triangle’s semiperimeter, whose area must be calculated. The semi-perimeter is equal to the number of the triangle’s three sides separated by two.

S= (a+b+c)/2

The three sides of the respective triangle are a, b, and c.

Finding the Area of Quadrilateral

If we know the lengths of the sides of a quadrilateral and any one of the diagonal lengths, we can divide the quadrilateral into two triangles and find the area for each using Heron’s formula by taking the diagonal as the common side. Finally, the areas of the two triangles must be added.


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