In geometry, Heron's formula (or Hero's formula) gives the area of a triangle in terms of the three side lengths Letting be the semiperimeter of the triangle,
s=\tfrac12(a+b+c),
A=\sqrt{s(s-a)(s-b)(s-c)}.
It is named after first-century engineer Heron of Alexandria (or Hero) who proved it in his work Metrica, though it was probably known centuries earlier.
Let be the triangle with sides
a=4,
b=13,
c=15.
s=\tfrac12(a+b+c)={}
\tfrac12(4+13+15)=16
\begin{align} A&=\sqrt{s(s-a)(s-b)(s-c)}=\sqrt{16 ⋅ (16-4) ⋅ (16-13) ⋅ (16-15)}\\ &=\sqrt{16 ⋅ 12 ⋅ 3 ⋅ 1}=\sqrt{576}=24. \end{align}
In this example, the side lengths and area are integers, making it a Heronian triangle. However, Heron's formula works equally well in cases where one or more of the side lengths are not integers.
Heron's formula can also be written in terms of just the side lengths instead of using the semiperimeter, in several ways,
\begin{align} A&=\tfrac{1}{4}\sqrt{(a+b+c)(-a+b+c)(a-b+c)(a+b-c)}\\[6mu] &=\tfrac{1}{4}\sqrt{2(a2b2+a2c2+b2c2)-(a4+b4+c4)}\\[6mu] &=\tfrac{1}{4}\sqrt{(a2+b2+c2)2-2(a4+b4+c4)}\\[6mu] &=\tfrac{1}{4}\sqrt{4(a2b2+a2c2+b2c2)-(a2+b2+c2)2}\\[6mu] &=\tfrac{1}{4}\sqrt{4a2b2-(a2+b2-c2)2}. \end{align}
After expansion, the expression under the square root is a quadratic polynomial of the squared side lengths,, .
The same relation can be expressed using the Cayley–Menger determinant,[2]
-16A2=\begin{vmatrix} 0&a2&b2&1\\ a2&0&c2&1\\ b2&c2&0&1\\ 1&1&1&0 \end{vmatrix}.
The formula is credited to Heron (or Hero) of Alexandria (60 AD),[3] and a proof can be found in his book Metrica. Mathematical historian Thomas Heath suggested that Archimedes knew the formula over two centuries earlier,[4] and since Metrica is a collection of the mathematical knowledge available in the ancient world, it is possible that the formula predates the reference given in that work.
A formula equivalent to Heron's, namely,
A=
1{2}\sqrt{a | |
2 |
c2-\left(
a2+c2-b2 | |
2 |
\right)2}
was discovered by the Chinese. It was published in Mathematical Treatise in Nine Sections (Qin Jiushao, 1247).[5]
There are many ways to prove Heron's formula, for example using trigonometry as below, or the incenter and one excircle of the triangle,[6] or as a special case of De Gua's theorem (for the particular case of acute triangles),[7] or as a special case of Brahmagupta's formula (for the case of a degenerate cyclic quadrilateral).
A modern proof, which uses algebra and is quite different from the one provided by Heron, follows.[8] Let be the sides of the triangle and the angles opposite those sides.Applying the law of cosines we get
\cos\gamma=
a2+b2-c2 | |
2ab |
From this proof, we get the algebraic statement that
\sin\gamma=\sqrt{1-\cos2\gamma}=
\sqrt{4a2b2-(a2+b2-c2)2 | |
The altitude of the triangle on base has length, and it follows
\begin{align} A &=\tfrac12(base)(altitude)\\[6mu] &=\tfrac12ab\sin\gamma\\[6mu] &=
ab | |
4ab |
\sqrt{4a2b2-(a2+b2-c2)2}\\[6mu] &=\tfrac14\sqrt{-a4-b4-c4+2a2b2+2a2c2+2b2c2}\\[6mu] &=\tfrac14\sqrt{(a+b+c)(-a+b+c)(a-b+c)(a+b-c)}\\[6mu] &=\sqrt{ \left(
a+b+c | |
2 |
\right) \left(
-a+b+c | |
2 |
\right) \left(
a-b+c | |
2 |
\right) \left(
a+b-c | |
2 |
\right)}\\[6mu] &=\sqrt{s(s-a)(s-b)(s-c)}. \end{align}
The following proof is very similar to one given by Raifaizen.[9] By the Pythagorean theorem we have
b2=h2+d2
a2=h2+(c-d)2
a2-b2=c2-2cd.
d=
-a2+b2+c2 | |
2c |
.
h2=b2-d2.
\begin{align} h2&=
| ||||
b |
\right)2\\ &=
(2bc-a2+b2+c2)(2bc+a2-b2-c2) | |
4c2 |
\\ &=
((b+c)2-a2)(a2-(b-c)2) | |
4c2 |
\\ &=
(b+c-a)(b+c+a)(a+b-c)(a-b+c) | |
4c2 |
\\ &=
2(s-a) ⋅ 2s ⋅ 2(s-c) ⋅ 2(s-b) | |
4c2 |
\\ &=
4s(s-a)(s-b)(s-c) | |
c2 |
. \end{align}
We now apply this result to the formula that calculates the area of a triangle from its height:
\begin{align} A&=
ch | |
2 |
\\ &=\sqrt{
c2 | |
4 |
⋅
4s(s-a)(s-b)(s-c) | |
c2 |
If is the radius of the incircle of the triangle, then the triangle can be broken into three triangles of equal altitude and bases and Their combined area is
A=\tfrac12ar+\tfrac12br+\tfrac12cr=rs,
s=\tfrac12(a+b+c)
The triangle can alternately be broken into six triangles (in congruent pairs) of altitude and bases and of combined area (see law of cotangents)
\begin{align} A&=r(s-a)+r(s-b)+r(s-c)\\[2mu] &=
| ||||
r |
+
s-b | |
r |
+
s-c | |
r |
\right)\\[2mu] &=
| ||||
r |
The middle step above is
\cot{\tfrac{\alpha}{2}}\cot{\tfrac{\beta}{2}}\cot{\tfrac{\gamma}{2}},
Combining the two, we get
A2=s(s-a)(s-b)(s-c),
Heron's formula as given above is numerically unstable for triangles with a very small angle when using floating-point arithmetic. A stable alternative involves arranging the lengths of the sides so that
a\geb\gec
A=\tfrac14\sqrt{(a+(b+c))(c-(a-b))(c+(a-b))(a+(b-c))}.
Three other formulae for the area of a general triangle have a similar structure as Heron's formula, expressed in terms of different variables.
First, if and are the medians from sides and respectively, and their semi-sum is
\sigma=\tfrac12(ma+mb+mc),
A=
4 | |
3 |
\sqrt{\sigma(\sigma-ma)(\sigma-mb)(\sigma-mc)}.
Next, if,, and are the altitudes from sides and respectively, and semi-sum of their reciprocals is
H=
-1 | |
\tfrac12l(h | |
a |
+
-1 | |
h | |
b |
+
-1 | |
h | |
c |
r),
A-1=4
-1 | |
\sqrt{Hl(H-h | |
a |
-1 | |
r)l(H-h | |
b |
-1 | |
r)l(H-h | |
c |
r)}.
Finally, if and are the three angle measures of the triangle, and the semi-sum of their sines is
S=\tfrac12(\sin\alpha+\sin\beta+\sin\gamma),
\begin{align} A&=D2\sqrt{S(S-\sin\alpha)(S-\sin\beta)(S-\sin\gamma)}\\[5mu] &=\tfrac12D2\sin\alpha\sin\beta\sin\gamma, \end{align}
where is the diameter of the circumcircle,
D=a/{\sin\alpha}=b/{\sin\beta}=c/{\sin\gamma}.
Heron's formula is a special case of Brahmagupta's formula for the area of a cyclic quadrilateral. Heron's formula and Brahmagupta's formula are both special cases of Bretschneider's formula for the area of a quadrilateral. Heron's formula can be obtained from Brahmagupta's formula or Bretschneider's formula by setting one of the sides of the quadrilateral to zero.Brahmagupta's formula gives the area of a cyclic quadrilateral whose sides have lengths as
K=\sqrt{(s-a)(s-b)(s-c)(s-d)}
where
s=\tfrac12(a+b+c+d)
Heron's formula is also a special case of the formula for the area of a trapezoid or trapezium based only on its sides. Heron's formula is obtained by setting the smaller parallel side to zero.
Expressing Heron's formula with a Cayley–Menger determinant in terms of the squares of the distances between the three given vertices,
A=
1 | |
4 |
\sqrt{-\begin{vmatrix} 0&a2&b2&1\\ a2&0&c2&1\\ b2&c2&0&1\\ 1&1&1&0 \end{vmatrix}}
Another generalization of Heron's formula to pentagons and hexagons inscribed in a circle was discovered by David P. Robbins.[16]
If are lengths of edges of the tetrahedron (first three form a triangle; opposite to and so on), then[17]
volume=
\sqrt{(-a+b+c+d)(a-b+c+d)(a+b-c+d)(a+b+c-d) | |
\begin{align} a&=\sqrt{xYZ}\\ b&=\sqrt{yZX}\\ c&=\sqrt{zXY}\\ d&=\sqrt{xyz}\\ X&=(w-U+v)(U+v+w)\\ x&=(U-v+w)(v-w+U)\\ Y&=(u-V+w)(V+w+u)\\ y&=(V-w+u)(w-u+V)\\ Z&=(v-W+u)(W+u+v)\\ z&=(W-u+v)(u-v+W). \end{align}
There are also formulae for the area of a triangle in terms of its side lengths for triangles in the sphere or the hyperbolic plane. [18] For a triangle in the sphere with side lengths and the semiperimeter
s=\tfrac12(a+b+c)
\tan2
S | |
4 |
=\tan
s2 | \tan | ||
|
s-b | \tan | |
2 |
s-c | |
2 |
\tan2
S | |
4 |
=\tanh
s2 | \tanh | ||
|
s-b | \tanh | |
2 |
s-c | |
2. |