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Difficulty: MediumAnalogy and Classification

Arrange the following geometric figures in a logical classification sequence based on the sum of their interior angles, in ascending order (from the smallest sum to the largest sum).

  1. 1Triangle
  2. 2Pentagon
  3. 3Hexagon
  4. 4Heptagon
  5. 5Octagon

Answer

The correct logical sequence from smallest to largest sum of interior angles is: Triangle, Pentagon, Hexagon, Heptagon, and Octagon.
The sum of the interior angles of a polygon increases as the number of its sides increases, defined by the formula S=(n−2)×180∘S = (n-2) \times 180^\circ. Therefore, ordering the shapes by their interior angle sums is mathematically identical to sorting them by their side count. The correct progression is Triangle (3 sides), Pentagon (5 sides), Hexagon (6 sides), Heptagon (7 sides), and Octagon (8 sides).

Step-by-Step Solution

1
Identify the relationship between a polygon and the sum of its interior angles.
The formula for the sum of interior angles is S=(n−2)×180∘S = (n-2) \times 180^\circ, where nn is the number of sides.
Establishing the mathematical rule is required to find the underlying numerical logic for the sequence.
2
Determine the number of sides (nn) for each geometric figure.
Triangle (n=3n=3), Pentagon (n=5n=5), Hexagon (n=6n=6), Heptagon (n=7n=7), Octagon (n=8n=8).
The number of sides directly dictates the magnitude of the interior angle sum.
3
Arrange the figures based on the side count in ascending order.
Since a larger number of sides always results in a larger sum of interior angles, ordering by nn gives: Triangle (3) < Pentagon (5) < Hexagon (6) < Heptagon (7) < Octagon (8).
This establishes the correct classification sequence from smallest to largest without needing to manually calculate every angle sum.

Key Concept

Classification based on numerical and geometric properties
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Topics

Analogy and Classification

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