Polygons • Topic 2 of 3

Regular Polygons

A regular polygon has all sides equal AND all angles equal — equilateral triangle, square, regular pentagon, hexagon, octagon. Because every angle is identical, each interior angle is just the total divided by n: (n−2)×180°/n, and each exterior angle is the fixed 360° shared equally, i.e. 360°/n. In CAT the most efficient route is almost always through the exterior angle, since it is small and divides 360° cleanly: a regular hexagon has 360/6 = 60° exterior and 120° interior; a regular octagon has 45° exterior and 135° interior. Two useful sanity checks: the interior angle of a regular polygon is always strictly less than 180° and rises towards 180° as n grows, and n must be a whole number — so if 360°/(180°−interior) is not an integer, the polygon described cannot be regular.

✅ Solved examples

1. Find each interior angle of a regular octagon.
Exterior = 360°/8 = 45°, so interior = 180° − 45° = 135°. (Check: (8−2)×180/8 = 1080/8 = 135°.)
2. Each interior angle of a regular polygon is 144°. Find the number of sides.
Exterior = 180° − 144° = 36°. n = 360°/36° = 10 sides (a regular decagon).
3. The interior angle of a regular polygon exceeds its exterior angle by 132°. How many sides?
Interior + exterior = 180° and interior − exterior = 132°. Adding: 2 × interior = 312° ⇒ exterior = 180° − 156° = 24°. n = 360/24 = 15.
4. Is a regular polygon with each interior angle 140° possible? If so, how many sides?
Exterior = 40°. n = 360/40 = 9, an integer, so yes — a regular nonagon.

✏️ Practice — try these, take hints as needed

1. Find each interior angle of a regular hexagon.
Exterior = 360/6.
Exterior = 60°.
Interior = 180 − 60.
120°
2. Each interior angle of a regular polygon is 108°. Find the number of sides.
Exterior = 180 − 108.
Exterior = 72°.
360 / 72.
5 sides (regular pentagon)
3. The ratio of an interior angle to its exterior angle in a regular polygon is 7 : 2. Find the number of sides.
They sum to 180°.
Exterior = (2/9) × 180 = 40°.
360 / 40.
9 sides
4. Each interior angle of a regular polygon is 165°. Find the number of sides.
Exterior = 180 − 165.
Exterior = 15°.
360 / 15.
24 sides
5. Can a regular polygon have each interior angle equal to 155°?
Exterior = 180 − 155 = 25°.
n = 360 / 25.
Is the result a whole number?
No (360/25 = 14.4 is not an integer)

📝 Topic test — 8 questions

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