Heron's formula for the area of a triangle
Named for the Greek Mathematician, Heron, who lived circa AD 62. He was
also called Hero.
Consider triangle ABC with sides of length a, b, and c:
If you let b be the base, then the height of the triangle is a sin(C), so the
area, K, of the triangle is given by:
K = (1/2)ab sin(C), where a and b are any two sides, and C is the included
angle.
The law of cosines tells us that c² = a² + b² - 2ab cos(C)
Solving for cos(C), we get cos(C) = (a² + b² - c²)/(2ab)
Since sin²(C) + cos²(C) = 1, we get this expression for sin(C):
sin(C) = sqrt(1 - ((a² + b² - c²) / (2ab) )²)
Plugging this into the area formula, above,
K = (1/2)ab sqrt(1 - ((a² + b² - c²) / (2ab) )²)
K = (1/4)(2ab) sqrt(1 - ((a² + b² - c²) / (2ab) )²)
K = (1/4) sqrt((2ab)² - (a² + b² - c²)²)
K = (1/4) sqrt(((2ab) + (a² + b² - c²))((2ab) - (a² + b² - c²)))
K = (1/4) sqrt((a² + 2ab + b² - c²)(-a² + 2ab - b² + c²))
K = (1/4) sqrt(((a+b)² - c²)(-(a-b)² + c²))
K = (1/4) sqrt((a+b+c)(a+b-c)(a-b+c)(-a+b+c))
K = sqrt(s(s-c)(s-b)(s-a)), where s=(a+b+c)/2, the semiperimeter
Internet References:
jwilson.coe.uga.edu/emt725/Heron/Heron.html
proves Heron's formula by expressing the height, h, of a triangle in terms of
its sides a, b, and c, and then using the area formula K = (1/2) ch to derive
the result.
mathforum.org/library/drmath/view/54957.html
proves it using K = b c sin(A), and then using needlessly complicated trig
substitutions to express sin(A) in terms of a, b, and c. Sort of like my
proof, except trickier. It reveals some interesting factoids along the
way, though.
Related pages in this website:
Summary of geometrical theorems
Law of Cosines
Point and Triangle -- a series
of pages explaining how to determine whether a point is inside a
triangle. One of these pages
gives an easy method using Heron's formula: Point D is inside triangle ABC if
the sums of the areas of ABD, BCD and CAD equal the area of ABC.
Heron's Formula for the
area of a triangle.
Brahmagupta's Formula
for the area of a cyclic quadrilateral, which can be considered a
generalization of Heron's Formula.
Bretschneider's Formula for the area of a quadrilateral,
which can be considered a generalization of Brahmagupta's Formula.
Triangle Area using Determinant
Triangle Area using Vectors
Polygon Area using Determinant
