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🔷 Chapter 12: Quadrilaterals • Exercise Set 12.4

Verified step-by-step solutions for your class notes.

Q1

Justify why the plane cannot be tiled with a regular pentagon.

Answer: Tiling the plane requires shapes to fit together around a vertex such that the sum of the angles meeting at that vertex equals exactly 360°, leaving no gaps or overlaps.
The interior angle of a regular pentagon is 108°.
If we try to fit regular pentagons around a vertex, three pentagons give 108° × 3 = 324°. This leaves a gap of 36°. A fourth pentagon would make the sum 432°, causing an overlap. Because 360 is not evenly divisible by 108, the plane cannot be tiled with regular pentagons.

108° 108° 108° 36° Gap! 3 × 108° = 324° < 360° (Cannot Tile)
Three Regular Pentagons at a Vertex Leave a 36° Uncovered Gap (4 × 108° = 432° > 360° Overlaps)