Count the number of sides in each shape in the sequence of Regular Polygons. Which number sequence do you get? What about the number of corners in each shape in the sequence of Regular Polygons? Do you get the same number sequence? Can you explain why this happens?
Counting the number of sides in Regular Polygons: Triangle has 3 sides Quadrilateral has 4 sides Pentagon has 5 sides Hexagon has 6 sides Heptagon has 7 sides Octagon has 8 sides Nonagon has 9 sides Decagon has 10 sides
Sequence obtained: 3, 4, 5, 6, 7, 8, 9, 10, ... This is the Counting numbers starting from 3.
Counting the number of corners (vertices): This gives the same sequence: 3, 4, 5, 6, 7, 8, 9, 10, ...
Answer: Yes, we get the same number sequence.
Explanation: Every polygon has the same number of corners as sides. Each side connects two corners, and each corner is where two sides meet. In any polygon, the number of vertices (corners) always equals the number of edges (sides).
Explanation
The chapter explains in section 1.6 that the number of sides in the Regular Polygons sequence follows counting numbers starting at 3. A regular polygon has equal-length sides and equal angles. The number of corners equals the number of sides because each side has two endpoints (corners), and each corner is shared by exactly two sides, creating a one-to-one correspondence between sides and corners.
Solution Steps
Count the number of sides in each Regular Polygon: Triangle (3), Quadrilateral (4), Pentagon (5), Hexagon (6), Heptagon (7), Octagon (8), Nonagon (9), Decagon (10).
Identify the sequence: 3, 4, 5, 6, 7, 8, 9, 10, ... which is the Counting numbers starting from 3.
Count the number of corners (vertices) in each Regular Polygon: same sequence 3, 4, 5, 6, 7, 8, 9, 10, ...
Compare both sequences — they are identical.
Explain why: In any polygon, the number of vertices equals the number of edges because each side connects two corners and each corner is shared by two sides.