Find the sum of all the interior angles of a pentagon.

For any simple polygon with n sides, the sum of its interior angles is given by the formula (n − 2) × 180°. This holds because any simple polygon can be triangulated into exactly (n − 2) non-overlapping triangles (for a convex polygon this triangulation can be done by drawing diagonals from a single vertex), and each triangle's angles sum to 180°.

1. A pentagon has 5 sides, so n = 5.

2. Substitute n = 5 into the formula: sum = (5 − 2) × 180°.

3. Simplify the bracket: 5 − 2 = 3, so sum = 3 × 180°.

4. Multiply: 3 × 180° = 540°.

As an independent check: a pentagon splits into exactly 3 triangles when diagonals are drawn from one vertex (5 − 2 = 3), and 3 × 180° = 540°, matching the result. A second check uses the exterior-angle rule: the exterior angles of any convex polygon always add up to 360°, and each interior angle is supplementary to its own exterior angle, so the interior-angle sum for a 5-sided figure is (5 × 180°) − 360° = 900° − 360° = 540° — the same value.

Hence, the sum of all interior angles of a pentagon is 540°.