Triangles: Angles, Sides, and Balance

Why triangles matter

A triangle is the simplest closed polygon: three sides, three angles, one region. That simplicity makes it the skeleton of roofs, bridges, and meshes—once three lengths lock, the shape cannot flex without stretching a side.

Angle sum

In Euclidean geometry, the interior angles of any triangle add to a straight angle:

α + β + γ = 180° = π radians

If you know two angles, the third is fixed. That single constraint ties local corner measures to the global shape.

Side relationships

Label the sides opposite angles A, B, and C as a, b, and c. Familiar special cases:

• Equilateral — all sides equal, all angles 60°
• Isosceles — two sides equal, base angles equal
• Right — one angle is 90°, and Pythagoras holds:

a² + b² = c²

Area without height drama

When height is awkward to measure, Heron’s formula still works from the three sides alone. With semi-perimeter s = (a + b + c) / 2:

Area = √[s(s - a)(s - b)(s - c)]

Takeaway

Triangles package constraint: angles sum to π, sides decide area and rigidity. Start here, and every later shape is triangles glued together.