Questions: Polygon Angle Sums

Question 1

A regular polygon has interior angles that each measure 150°. How many sides does it have?

• A. 10 sides
• B. 12 sides
• C. 8 sides
• D. 15 sides

Answer: (B)

Each interior angle is 150°, so each exterior angle is 180° − 150° = 30° (since interior and exterior angles at each vertex are supplementary). The exterior angle sum of any convex polygon is always 360°, so the number of sides = 360° ÷ 30° = 12. You can verify with the interior formula: (n−2)×180° ÷ n = 150° → (n−2)×180 = 150n → 180n − 360 = 150n → 30n = 360 → n = 12. The exterior angle shortcut is often faster.

Question 2

How does the sum of interior angles change when a polygon gains one additional side — for example, going from a pentagon to a hexagon?

• A. It stays the same — the interior angle sum formula always produces 360° regardless of n
• B. It increases by 180° — each additional side adds exactly one more triangle to the triangulation
• C. It doubles — a hexagon has twice the interior angle sum of a triangle
• D. It increases by 360° — because the exterior angle sum is always 360°, the interior sum grows by the same amount

Answer: (B)

Each additional side adds exactly one more triangle to the triangulation from a fixed vertex: a pentagon splits into 3 triangles (sum = 540°), a hexagon into 4 (sum = 720°). The difference is always 180°. Formally: (6−2)×180° − (5−2)×180° = 4×180° − 3×180° = 180°. Every additional side contributes exactly one triangle's worth of angle to the total. This is why the formula is (n−2)×180° and not n×180°.

Question 3

The sum of the exterior angles of a regular hexagon is greater than the sum of the exterior angles of a regular triangle, because the hexagon has more sides and therefore more exterior angles.

• T. True
• F. False

Answer: False

The sum of exterior angles of any convex polygon is always exactly 360°, regardless of the number of sides. A triangle has three exterior angles of 120° each: 3 × 120° = 360°. A hexagon has six exterior angles of 60° each: 6 × 60° = 360°. More sides means each individual exterior angle is smaller, but there are more of them — the total is invariant at 360°. This constant exterior angle sum is what makes the walking-the-perimeter intuition work.

Question 4

Every interior angle of a regular polygon with n sides measures (n−2) × 180° ÷ n degrees.

• T. True
• F. False

Answer: True

In a regular polygon, all interior angles are equal. The total interior angle sum is (n−2)×180°, and dividing equally among n angles gives (n−2)×180° ÷ n per angle. For a regular hexagon: 4×180° ÷ 6 = 720° ÷ 6 = 120°. This is why hexagonal tiles fit together perfectly at vertices — three 120° angles sum to exactly 360°, filling the space around a point without gaps or overlap. For a regular triangle: 1×180° ÷ 3 = 60° per angle, as expected.

Question 5

Why does the sum of exterior angles of any convex polygon always equal 360°, regardless of the number of sides?

Model answer: Imagine walking the perimeter of a convex polygon. At each vertex, you turn by the exterior angle — the supplement of the interior angle. After completing the full circuit and returning to your starting position facing the original direction, you have rotated exactly once — a total of 360°. This total turning is invariant because it equals one complete revolution, regardless of how many sides the polygon has or how the 360° is distributed across individual turns.

With more sides, each individual turn is smaller, but there are more of them — the total is always 360°. For a triangle, three large turns (120° each) sum to 360°. For a regular decagon, ten small turns (36° each) also sum to 360°. The insight is that the total angular change depends on the topological fact that you traversed a simple closed curve and returned to your starting orientation — not on the specific number of sides. This also connects back to the interior formula: since interior + exterior = 180° at each vertex, total interior + 360° = n×180°, so total interior = (n−2)×180°.