Quadrilaterals & Polygons • Topic 3 of 4

Interior & Exterior Angles

For any polygon with n sides, the interior angles sum to (n − 2) × 180° — split the shape into (n − 2) triangles to see why. In a regular polygon (all sides and angles equal) each interior angle is that sum divided by n. The cleaner tool for many GRE problems is the exterior angles: no matter how many sides, they always sum to 360°, so each exterior angle of a regular polygon is simply 360°/n. Interior and exterior angles at a vertex are supplementary, which lets you jump between them in one step. To find the number of sides from an angle, work through the exterior angle: sides = 360° / (exterior angle).

A regular pentagon with one interior angle and one exterior angle markedInterior and exterior angles108°72°Interior + exterior = 180°

✅ Solved examples

1. Find the sum of the interior angles of a hexagon (6 sides).
Sum = (n − 2) × 180° = (6 − 2) × 180° = 4 × 180° = 720°.
2. Find each interior angle of a regular pentagon.
Interior sum = (5 − 2) × 180° = 540°; each angle = 540° / 5 = 108°.
3. Find each exterior angle of a regular octagon (8 sides).
Exterior angles sum to 360°, so each = 360° / 8 = 45°.
4. A regular polygon has interior angles of 150°. How many sides does it have?
Each exterior angle = 180° − 150° = 30°. Sides = 360° / 30° = 12.

✏️ Practice — try these, take hints as needed

1. Find the sum of the interior angles of an octagon (8 sides).
(n − 2) × 180°.
(8 − 2) × 180.
6 × 180.
1080°
2. Find each interior angle of a regular hexagon.
Interior sum = 720°.
Divide by 6.
720 / 6.
120°
3. Find each exterior angle of a regular decagon (10 sides).
Exterior angles sum to 360°.
Divide by 10.
360 / 10.
36°
4. A regular polygon has exterior angles of 40°. How many sides does it have?
Sides = 360° / exterior angle.
360 / 40.
Count the sides.
9
5. A regular polygon has interior angles of 135°. How many sides does it have?
Exterior = 180 − 135.
Exterior = 45°.
Sides = 360 / 45.
8

📝 Topic test — 8 questions

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