Semiperimeter

Definition

Semiperimeter ($s$) is exactly half of the perimeter of a closed two-dimensional figure.

Why It Matters

The semiperimeter is the “hidden lever” that makes complex area calculations (like Heron’s formula) possible, simplifying equations in geometry and trigonometry.

Core Concepts

• Semiperimeter Formulas:
  • Triangles (Heron’s): $s = \frac{a+b+c}{2}$
  • Cyclic Quadrilaterals (Brahmagupta’s): $s = \frac{a+b+c+d}{2}$
  • How to read: “The semiperimeter s is equal to the sum of a, b, and c all divided by two; and the semiperimeter s is equal to the sum of a, b, c, and d all divided by two.”
  • Meaning / when to use: Half the perimeter—plug into Heron’s or Brahmagupta’s area formulas.

Connected Concepts

• Perimeter
• Area of a Triangle
• Area of Quadrilaterals
• Brahmagupta’s Formula