A rhombus has all four sides equal but its angles are not all equal. Is a rhombus a regular polygon?
AYes — a polygon only needs equal sides to be regular
BNo — a regular polygon requires both all sides equal AND all angles equal
CYes, but only if all angles are multiples of 90°
DNo — regular polygons must have more than four sides
A regular polygon must satisfy two conditions simultaneously: all sides congruent (equilateral) AND all angles congruent (equiangular). A rhombus is equilateral but not equiangular — its acute and obtuse angles differ — so it is not regular. This distinction matters for n > 3: only triangles are automatically equiangular when equilateral. For all other polygons, the two conditions are independent.
Question 2 Multiple Choice
Why can't regular pentagons tile the plane, while regular hexagons can?
APentagons have too many sides to fit together without gaps
BThe interior angle of a regular pentagon (108°) does not divide evenly into 360°, so pentagons cannot meet exactly at a vertex
CPentagons are not symmetric enough to form a tiling pattern
DRegular pentagons have exterior angles too large to allow edge-to-edge contact
For tiles to meet at a vertex without gaps or overlaps, their angles must sum to exactly 360°. A regular hexagon's interior angle is 120°, and 360 ÷ 120 = 3 — so exactly three hexagons meet at each vertex. A regular pentagon's interior angle is 108°, and 360 ÷ 108 is not a whole number (≈ 3.33), so pentagons cannot meet exactly at a vertex. Only equilateral triangles (60°, six per vertex), squares (90°, four per vertex), and regular hexagons (120°, three per vertex) satisfy this divisibility condition.
Question 3 True / False
The exterior angle of a regular polygon equals 360° divided by the number of sides.
TTrue
FFalse
Answer: True
This follows from a simple walking argument: if you traverse all edges of any convex polygon, you turn a full 360° total. For a regular polygon, every turn is equal, so each exterior angle is 360°/n. For example, a regular octagon (n = 8) has exterior angles of 45°, and 8 × 45° = 360°. This formula is actually simpler than the interior angle formula and makes geometric sense as n grows: as n increases, 360°/n approaches 0°, meaning the polygon approaches a circle with almost no turning at each vertex.
Question 4 True / False
Any polygon with most sides equal (equilateral) is also a regular polygon.
TTrue
FFalse
Answer: False
This is the most common misconception about regular polygons. A rhombus is equilateral — all four sides are equal — but its angles are not equal (it has two pairs of angles: acute and obtuse). So a rhombus is equilateral but NOT regular. For polygons with more than three sides, equilateral does not imply equiangular. Only for triangles is this automatically true: an equilateral triangle must also be equiangular (all 60°). For all other polygons, you must verify both conditions independently.
Question 5 Short Answer
Exactly three regular polygons can tile the plane by themselves. What determines whether a regular polygon can tile the plane, and why does this rule out most polygons?
Think about your answer, then reveal below.
Model answer: A regular polygon can tile the plane if and only if its interior angle divides evenly into 360°. At each vertex of a tiling, the angles of the meeting polygons must sum to exactly 360° with no gaps or overlaps. For equilateral triangles (60°): 360 ÷ 60 = 6 ✓. For squares (90°): 360 ÷ 90 = 4 ✓. For regular hexagons (120°): 360 ÷ 120 = 3 ✓. For regular pentagons (108°): 360 ÷ 108 ≈ 3.33 ✗. For regular heptagons and beyond, interior angles exceed 120°, so fewer than 3 could meet at a vertex, still not dividing 360° evenly. This divisibility test eliminates all other regular polygons.
The key insight is that tiling is a constraint on angles at vertices, not on edge lengths. Any regular polygon can have its edges matched (they're all equal), but unless its interior angle divides 360° evenly, vertices cannot close up without leaving a gap or forcing an overlap. This makes tiling a number-theoretic question disguised as a geometry problem.