Polygons with both equal sides and equal angles — and the exterior angle shortcut that makes them easy to work with.
The Mnemonic
Two Conditions, Both Required
A regular polygon requires BOTH conditions at once: equilateral (all sides equal length) AND equiangular (all angles equal measure). Having just one of these conditions isn't enough — a shape can be equilateral without being equiangular (like a rhombus that isn't a square), or equiangular without being equilateral (like a rectangle that isn't a square).
Regular polygons come with a genuinely useful shortcut: the sum of the EXTERIOR angles of ANY convex polygon — regular or not — is always exactly 360°. For a regular polygon specifically, dividing 360° by the number of sides gives you each individual exterior angle directly.
All sides equal, all angles equal — every regular polygon is both equilateral AND equiangular
💡 Memory Trick
"Polygons with both equal sides and equal angles." Regular = equilateral AND equiangular — both conditions, not just one. And remember the exterior angle shortcut works for EVERY convex polygon (360° total), while the interior angle sum formula (n-2)×180° also works for every polygon — regular polygons just let you divide either total evenly since every piece is identical.
Why It Works
Why Exterior Angles Always Sum to 360°
Picture walking around the entire perimeter of any convex polygon, turning at each vertex by exactly the exterior angle there. By the time you've walked all the way around and returned to your starting point facing the same direction you started, you've made one complete rotation — 360° total, no matter how many sides the polygon has or how large the individual angles are.
This is actually a much simpler and more universal fact than the interior angle sum formula — it doesn't even need the (n-2) adjustment, because it's tied directly to one full rotation rather than to how many triangles fit inside the shape.
Using It In A Proof
Choosing Interior vs. Exterior Angle Approach
For regular polygons specifically, the exterior angle approach is often the FASTER path to finding one interior angle, rather than going through the full (n-2)×180° formula and then dividing.
1
For a regular polygon, find one exterior angle first
Each exterior angle = 360° / n — a single division, no subtraction needed.
2
Convert to the interior angle using the linear pair relationship
Each interior angle = 180° − (each exterior angle), since interior and exterior angles at the same vertex form a linear pair.
3
Or work backward from a given angle to find n
If given one exterior angle, n = 360° / (exterior angle). If given one interior angle, first find the exterior angle (180° − interior), then divide 360° by that.
Full Worked Example
Finding the Number of Sides From an Interior Angle
Given: A regular polygon has an interior angle of 156°. Find: the number of sides.
1
Find the exterior angle
Since interior and exterior angles form a linear pair: exterior angle = 180° − 156° = 24°.
2
Use the 360° shortcut
n = 360° / (exterior angle) = 360° / 24°.
3
Calculate
n = 15 — this is a regular 15-sided polygon (a pentadecagon).
Compare this to solving via (n-2)×180° = n×156°, which requires more algebra to isolate n — the exterior angle approach is faster specifically because 360° is a fixed, simple total that doesn't depend on n at all.
🎯 Quick Worked Example
A regular polygon has 9 sides. Find its exterior and interior angle measures.
1
Find the exterior angle. 360° / 9 = 40°.
2
Find the interior angle. 180° − 40° = 140°.
3
Check with the other formula. (9−2)×180° = 1260°, and 1260°/9 = 140°. ✓ Matches.
📌 Exam Application
When a problem gives you one interior (or exterior) angle of a regular polygon and asks for the number of sides, the exterior-angle route (n = 360°/exterior angle) is almost always faster than setting up the full (n-2)×180° equation — reach for 360° first whenever a regular polygon's angle is given.
⚠️ Most Common Regular Polygons Mistakes
Trap 1 — Assuming equilateral alone means regular: A shape needs BOTH equal sides AND equal angles to be regular — a rhombus (equal sides, unequal angles in general) is a common example of equilateral without being regular.
Trap 2 — Using 360° for the INTERIOR angle sum: The 360° shortcut applies specifically to the sum of EXTERIOR angles — the interior angle sum still requires (n-2)×180°, a completely different (and n-dependent) total.
✓ Quick Self-Test
1) What two conditions must both be true for a polygon to be regular? 2) What is the sum of the exterior angles of ANY convex polygon? 3) A regular polygon has 12 sides — find its exterior and interior angle measures. 4) A regular polygon has an exterior angle of 20° — how many sides does it have? 5) Why is the exterior angle approach often faster than (n-2)×180° for finding one angle of a regular polygon?