Asked by Damianah Joy Locher
What is a Heptagon?
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Answered by
GPT-5 mini
AI
A heptagon (also called a septagon) is a polygon with seven sides and seven vertices.
Key facts
- Number of sides/vertices: 7
- Sum of interior angles: (7 − 2)·180° = 900°
- Each interior angle of a regular heptagon: 900° / 7 ≈ 128.571°
- Each exterior angle of a regular heptagon: 360° / 7 ≈ 51.429°
- Number of diagonals: 7·(7 − 3)/2 = 14
- Symmetry (regular): dihedral group D7, order 14 (7 rotations and 7 reflections)
- A regular heptagon is not constructible with straightedge and compass (7 is not a Fermat prime).
Formulas for a regular heptagon with side length a
- Area: A = (7/4) a^2 cot(π/7) ≈ 1.75 a^2 · 2.07768 ≈ 3.635 a^2
- Circumradius: R = a / (2 sin(π/7)) ≈ a · 1.15238
- Inradius: r = a / (2 tan(π/7)) ≈ a · 1.03826
If you want, I can show a diagram, give coordinates of the vertices on a circle, or show an approximate straightedge-and-compass construction.
Key facts
- Number of sides/vertices: 7
- Sum of interior angles: (7 − 2)·180° = 900°
- Each interior angle of a regular heptagon: 900° / 7 ≈ 128.571°
- Each exterior angle of a regular heptagon: 360° / 7 ≈ 51.429°
- Number of diagonals: 7·(7 − 3)/2 = 14
- Symmetry (regular): dihedral group D7, order 14 (7 rotations and 7 reflections)
- A regular heptagon is not constructible with straightedge and compass (7 is not a Fermat prime).
Formulas for a regular heptagon with side length a
- Area: A = (7/4) a^2 cot(π/7) ≈ 1.75 a^2 · 2.07768 ≈ 3.635 a^2
- Circumradius: R = a / (2 sin(π/7)) ≈ a · 1.15238
- Inradius: r = a / (2 tan(π/7)) ≈ a · 1.03826
If you want, I can show a diagram, give coordinates of the vertices on a circle, or show an approximate straightedge-and-compass construction.
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