Semiperimeter

The semiperimeter of a polygon is half its perimeter, found by adding its side lengths and dividing by two. It is commonly denoted by s in triangle geometry.

Heron's formula: A formula that calculates a triangle’s area from its three side lengths. It uses the semiperimeter to express area as √[s(s−a)(s−b)(s−c)].
Perimeter: The total length of a closed polygon’s boundary. Semiperimeter is exactly half this boundary length.
Polygon: A closed plane figure bounded by finitely many straight line segments. Semiperimeter applies to polygons of any number of sides, not just triangles.
Area: The measure of the two-dimensional region enclosed by a figure. Area measures interior extent, unlike semiperimeter’s boundary-based length.
Inradius: The radius of a polygon’s inscribed circle; for a triangle, it is the perpendicular distance from the incenter to a side. A triangle’s area equals its inradius multiplied by its semiperimeter.
Triangle inequality: The rule that each side of a triangle is shorter than the sum of its other two sides. It ensures each difference s−a, s−b, and s−c is positive for a nondegenerate triangle.
Triangle: A polygon with three sides and three angles. Triangle geometry is where the symbol s and many semiperimeter formulas are standard.
Radius: The distance from the center of a circle or sphere to its boundary. A radius is a center-to-boundary distance, while semiperimeter sums polygon edges and halves the result.
Exradius: The radius of a circle tangent to one side of a triangle and the extensions of its other two sides. Triangle exradii use the semiperimeter and the area in formulas parallel to the inradius formula.
Triangle side lengths: The three positive lengths of a triangle’s edges. Their sum determines the triangle’s semiperimeter.

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