Angles in Polygons

طریقہ، مرحلہ وار

1. 1

   Find the interior angle sum

   hexagon, n = 6 → (6 − 2) × 180 = 720°

   Splitting a polygon into triangles from one vertex always produces two fewer triangles than there are sides, and each triangle contributes 180°. The formula is that observation written down, not a fact to memorise blindly.

2. 2

   For a regular polygon, divide by n

   720 ÷ 6 = 120° per interior angle

   Regular means all sides and all angles are equal, so the total shares out evenly. This step only applies to regular polygons — an irregular hexagon still totals 720° but its individual angles differ.

3. 3

   Exterior angles always total 360°

   regular hexagon → 360 ÷ 6 = 60° each

   Walking once around any polygon turns you through a full circle, so the exterior angles sum to 360 regardless of the number of sides. This is often the faster route to an interior angle: 180 − 60 = 120.

4. 4

   Interior and exterior angles are supplementary

   interior + exterior = 180° at every vertex

   They sit on a straight line together, so knowing one immediately gives the other. Many questions are quicker approached through the exterior angle even when they ask about the interior one.

5. 5

   Work backwards to find the number of sides

   exterior angle 24° → n = 360 ÷ 24 = 15 sides

   Dividing 360 by the exterior angle gives the number of sides directly. If the question supplies the interior angle instead, subtract from 180 first — that conversion is where most errors in this question type occur.

Where (n − 2) × 180 comes from

Pick one vertex of a polygon and draw diagonals to every other vertex. A quadrilateral splits into 2 triangles, a pentagon into 3, a hexagon into 4 — always two fewer than the number of sides, because the two neighbouring vertices cannot be joined to. Each triangle contributes 180°.

Students who have drawn this once do not forget the formula, and more importantly they can rebuild it. A student who has only memorised it will sometimes write (n + 2) or n × 180 under pressure and have no way to notice.

• Interior sum = (n − 2) × 180°
• Regular polygon interior angle = sum ÷ n
• Exterior angles always sum to 360°
• Interior + exterior = 180° at each vertex
• n = 360 ÷ exterior angle

The exterior angle is usually the shortcut

Because exterior angles always total 360 no matter how many sides, questions about regular polygons are frequently one division away from an answer. Finding the interior angle of a regular 20-gon via the interior sum takes two steps and larger numbers; via the exterior angle it takes 360 ÷ 20 = 18, then 180 − 18 = 162.

This is worth teaching as the default approach for regular polygons, with the interior sum reserved for irregular ones where the exterior angles are not all equal.

Irregular polygons

The interior sum formula holds for every polygon, regular or not — an irregular pentagon still totals 540°. What fails for irregular shapes is dividing by n, since the angles are not equal, so questions give all but one angle and ask for the missing one.

The method there is to total the known angles and subtract from the sum. Students who reach for the divide-by-n step out of habit produce an answer that ignores the information the question actually gave.

How we teach polygon angles

We have students draw the triangle split themselves for three or four polygons before showing the formula. It takes ten minutes and converts (n − 2) × 180 from an arbitrary string into something they watched happen.

We then teach the exterior-angle route as the primary method for regular polygons, because it is fewer steps with smaller numbers and it works identically for a triangle and a fifty-sided shape.