Stage 9 - Q1 - 2 marks
Find one interior angle of a regular 6-gon.
Hint
Regular means all angles, as well as all sides, are equal.
Stage 9 - Q2 - 2 marks
A regular polygon has interior angle \(120^\circ\). Find its number of sides.
Hint
Use the angle needed to turn from one side to the next.
Stage 9 - Q3 - 3 marks
Can congruent regular 9-gons tile around a vertex with no gap or overlap? Justify using angles.
Hint
The number of polygons meeting at the point must be an integer.
Stage 9 - Q4 - 3 marks
The regular polygons shown share the marked vertex. Find the unfilled angle.
Hint
Use the regular interior angle formula for each polygon.
Stage 9 - Q5 - 4 marks
Two equilateral triangles and a square, each with side 9 mm, meet at a point. Which regular polygon could fill the remaining angle? Give its side count.
Hint
Find the missing interior angle before finding the number of sides.
Stage 9 - Q6 - 4 marks
Two equilateral triangles and a square, each with side 10 mm, meet at a point. Which regular polygon could fill the remaining angle? Give its side count.
Hint
Find the missing interior angle before finding the number of sides.
Stage 9 - Q7 - 5 marks
Draw a patch of a tessellation using regular 3-gons with side 4 cm. Show one completely filled central vertex and justify that it has no angular gap.
Hint
Use the known triangle, square or regular hexagon tessellation.
Stage 9 - Q8 - 5 marks
One regular 9-gon and two identical regular polygons fill a vertex. Determine whether the two unknown polygons are possible and find their side count if so.
Hint
The angle equation is necessary, and the side count must also be a whole number.
Stage 9 - Q9 - 5 marks
Find every possible number of sides of a regular polygon whose interior angle is strictly between these values.
Hint
Compare with the regular polygons at the two endpoint angles.
Stage 9 - Q10 - 3 marks
Only congruent squares and equilateral triangles are allowed around one vertex. Find every pair of nonnegative tile counts \((s,t)\) that fills the vertex. Which possibility uses both shapes?
Hint
The square count must be even, and it cannot exceed 4.