What is the measure of each interior angle in a regular eight-sided polygon?

Correct — B. A regular octagon has eight equal interior angles, so find their total and share it out.

Triangles from one vertex: n − 2 = 8 − 2 = 6
Angle total: 6 × 180° = 1080°
Each angle: 1080° ÷ 8 = 135°

Check with the exterior angle: 360° ÷ 8 = 45°, and 180° − 45° = 135° → option (b)

Why the others are wrong

• (a)
  120° — 120° is the interior angle of a regular hexagon: (6 − 2) × 180° ÷ 6 = 120°. An octagon has two more sides, so each of its angles is wider.
• (c)
  150° — 150° belongs to a regular 12-sided polygon, whose exterior angle is 360° ÷ 12 = 30°. The octagon's exterior angle is 45°, which leaves 135°.
• (d)
  165° — 165° needs an exterior angle of only 15°, which means 360° ÷ 15 = 24 sides. With 8 sides the exterior angle is 45°, not 15°.

Any polygon with n sides splits into n − 2 triangles when you draw diagonals from one vertex, so its interior angles total (n − 2) × 180°.

In a regular polygon every angle is equal, so one angle is that total ÷ n. The quicker route uses the exterior angles, which total 360°: each is 360° ÷ n, and the interior angle is 180° minus it.

Key facts

• Sum of the interior angles of an n-sided polygon = (n − 2) × 180°.
• The exterior angles of a convex polygon total 360°, so a regular polygon's exterior angle is 360° ÷ n.
• Regular octagon: angle total 1080°, each interior angle 135°, each exterior angle 45°.

Common traps

• Stopping at 360° ÷ 8 = 45°: that is the exterior angle, and the interior angle is 180° − 45°.
• Giving the total, 1080°, when the question asks for each angle.

No directly related past PYQ was found.