The sum of the exterior of an n-sided convex polygon is half the sum of its interior angle. find n

Analyze the question: We have a convex polygon with n sides. The sum of its exterior angles is half the sum of its interior angles. We need n.

Concept:

• The sum of the exterior angles of any convex polygon = 360°
• The sum of the interior angles of an n-sided polygon = (n − 2) × 180°

Solution:

Exterior sum = ½ × Interior sum

360 = ½ × (n − 2) × 180

360 = 90(n − 2)

n − 2 = 360 ÷ 90 = 4

n = 6

Check: A hexagon has interior angle sum = (6 − 2) × 180 = 720°. Half of 720° = 360°, which equals the exterior angle sum. ✔

Tip: The exterior angle sum is always 360° for any convex polygon, no matter how many sides it has. So whenever a question compares the exterior sum to the interior sum, set 360 equal to the relationship and solve for (n − 2) × 180. Also remember: a quick check is that the interior sum is 720° for a hexagon, which is exactly double 360°.