Sum of Interior Angles

Definition

The sum of interior angles in any polygon is determined by the number of sides $n$, following the formula $\Sigma \text{ interior angles} = (n - 2) \times 180^\circ$.

• How to read: “Sum of interior angles equals (n minus two) times one hundred eighty degrees.”
• Meaning: An $n$-gon can be triangulated into $n - 2$ triangles, each contributing $180^\circ$.

Why It Matters

This formula provides a “geometric invariant” for any polygon, enabling error-checking in engineering and surveying; it reveals that regardless of how a shape is distorted, its internal “turning” is strictly governed by its number of sides.

Core Concepts

• Interior Angle Sum: The sum $S$ of interior angles for an $n$-sided polygon is: $S = (n - 2) \cdot 180^\circ$
  • How to read: “S equals (n minus two) times one hundred eighty degrees.”
  • Meaning: The total turning inside the polygon equals the sum of angles in $n - 2$ non-overlapping triangles.
• Exterior Angle Sum: The sum of the exterior angles (one at each vertex) of any convex polygon is always $360^\circ$.
  • How to read: “Exterior angles always sum to three hundred sixty degrees.”
  • Meaning: Walking once around the perimeter, you make one full rotation regardless of the number of sides.
• Regular Polygons (Equiangular):
  • Each Interior Angle ($I$): $I = \frac{(n - 2) \cdot 180^\circ}{n}$
    • How to read: “I equals (n minus two) times one eighty, over n.”
    • Meaning: Divide the total interior sum equally among all $n$ equal angles.
  • Each Exterior Angle ($E$): $E = \frac{360^\circ}{n}$
    • How to read: “E equals three sixty over n.”
    • Meaning: Each exterior turn is an equal share of one full revolution.
• Diagonal Count: The number of unique diagonals $D$ in a polygon is: $D = \frac{n(n - 3)}{2}$
  • How to read: “D equals n times (n minus three) over two.”
  • Meaning: Each vertex connects to $n - 3$ non-adjacent vertices; divide by 2 because each diagonal is counted twice.
• Triangles ($n=3$): Interior sum is $180^\circ$.
  • How to read: “For n equals three, interior sum is one eighty.”
  • Meaning: The base case: one triangle, one $180^\circ$ sum.
• Quadrilaterals ($n=4$): Interior sum is $360^\circ$.
  • How to read: “For n equals four, interior sum is three sixty.”
  • Meaning: Two triangles fit inside any quadrilateral, giving $(4-2) \times 180^\circ = 360^\circ$.

Connected Concepts

• Triangle Classification
• Degree Measure