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The area A of any triangle is the product of its inradius (the radius of its inscribed circle) and its semiperimeter: A = r s . {\displaystyle A=rs.} The area of a triangle can also be calculated from its semiperimeter and side lengths a, b, c using Heron's formula :
A triangle with sides a, b, and c. 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, = (+ +), the area is [1]
The area of a triangle can be demonstrated, for example by means of the congruence of triangles, ... is the semiperimeter, or half of the triangle's perimeter.
A triangle may be regarded as a quadrilateral with one side of length zero. From this perspective, as d approaches zero, a cyclic quadrilateral converges into a cyclic triangle (all triangles are cyclic), and Brahmagupta's formula simplifies to Heron's formula. If the semiperimeter is not used, Brahmagupta's formula is
where , , and are the area, radius of the incircle, and semiperimeter of the original triangle, and , , and are the side lengths of the original triangle. This is the same area as that of the extouch triangle. [20]
The parameters most commonly appearing in triangle inequalities are: the side lengths a, b, and c;; the semiperimeter s = (a + b + c) / 2 (half the perimeter p);; the angle measures A, B, and C of the angles of the vertices opposite the respective sides a, b, and c (with the vertices denoted with the same symbols as their angle measures);
A man who was running away from police has been arrested after getting stuck in a chimney while trying to hide from them, authorities said. The incident occurred on Tuesday evening in Fall River ...
The semiperimeter s of a Heronian triangle cannot be prime (as () () is the square of the area, and the area is an integer, if s were prime, it would divide another factor; this is impossible as these factors are all less than s).