Heron's formula

Heron's formula is a special way to find the area of a triangle when you know the lengths of all three sides. This formula was created by Heron of Alexandria, an engineer who lived over 2,000 years ago. He wrote about it in his work called Metrica.

To use the formula, you first need to find something called the semiperimeter. This is just half the total length around the triangle. Once you have the semiperimeter, you can use a simple calculation to find the area. This method works for any triangle, no matter its shape or size.

Heron's formula helps us solve many problems in geometry. It shows how math can connect different parts of a shape in surprising ways. Even though Heron lived so long ago, his idea is still useful today for people who study shapes and sizes.

Example

Imagine a triangle with sides measuring 4, 13, and 15 units. To find its area using Heron's formula, we first add these sides together and divide by 2, giving us 16. Then we subtract each side from this number: 16 minus 4 is 12, 16 minus 13 is 3, and 16 minus 15 is 1. By multiplying these results together and taking the square root, we find the area to be 24 square units.

Heron's formula can be used for any triangle, even if its sides are not whole numbers, as long as the sides follow the triangle rule. This means that the sum of any two sides must always be greater than the third side.

Alternative expressions

Heron's formula can also be written in different ways using just the side lengths of the triangle, instead of using the semiperimeter. These formulas still give the same result for the area of the triangle.

After expanding, the part under the square root becomes a special kind of math expression called a quadratic polynomial, which uses the squares of the side lengths.

The same idea can also be shown using something called the Cayley–Menger determinant.

History

Heron's formula is named after Heron of Alexandria, who lived around 60 AD. He wrote about it in his book called Metrica. Some experts think that a famous mathematician named Archimedes may have known about the formula even earlier.

A similar way to find the area of a triangle was also found by a Chinese mathematician named Qin Jiushao. He shared his method in a book called Mathematical Treatise in Nine Sections in the year 1247.

Proofs

There are many ways to prove Heron's formula. One way uses trigonometry, another uses the incenter and one excircle of the triangle, and yet another is a special case of De Gua's theorem or Brahmagupta's formula.

Trigonometric proof using the law of cosines

A modern proof uses algebra and the law of cosines. Let a, b, and c be the sides of the triangle and α, β, and γ the angles opposite those sides. By using the law of cosines, we can find the area of the triangle.

Algebraic proof using the Pythagorean theorem

Another proof uses the Pythagorean theorem. By this theorem, we can express certain lengths in terms of the triangle's sides. This helps us find the area of the triangle.

Trigonometric proof using the law of cotangents

We can also use the radius of the incircle of the triangle. By breaking the triangle into smaller triangles, we can find the area using the semiperimeter and other values.

Numerical stability

Heron's formula can sometimes give incorrect results for triangles with very small angles. This happens because of small errors that occur when doing calculations with limited numbers, like on a computer.

One way to avoid this problem is to arrange the side lengths so that the longest side comes first, followed by the next longest, and then the shortest. Using this order helps make the calculations more accurate.

Similar triangle-area formulae

There are three other ways to find the area of a triangle, similar to Heron’s formula. These use different parts of the triangle.

One way uses the triangle’s medians—the lines from each corner to the middle of the opposite side. If you know these medians, you can find the area using a special formula.

Another way uses the triangle’s heights, called altitudes. These are the distances from each corner straight down to the opposite side. There is a formula that uses these heights to find the area.

The last way uses the triangle’s angles and the size of the circle that passes through all three corners. With these, you can also find the area using another special formula.

Generalizations

Heron's formula is a special case of Brahmagupta's formula for the area of a cyclic quadrilateral. Both formulas are special cases of Bretschneider's formula for the area of a quadrilateral. Heron's formula can be obtained by setting one side of the quadrilateral to zero.

Brahmagupta's formula gives the area of a cyclic quadrilateral based on its four side lengths. Heron's formula is also related to the area of a trapezoid when one of its parallel sides is set to zero.

Heron's formula can also be expressed using a Cayley–Menger determinant, showing its connection to other geometric formulas. There are also generalizations of Heron's formula for shapes like pentagons and hexagons inscribed in a circle.