Sum of Interior Angles of a Polygon

Overview

An animation that visually derives the formula for the sum of interior angles of an $n$-sided polygon, $S = (n-2)\times180^{\circ}$, using triangulation and shows examples for triangles, quadrilaterals, and pentagons. The key takeaway is how each additional side adds another triangle, increasing the total interior angle sum by $180^{\circ}$.

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Phases

#	Phase Name	Duration	Description
1	Intro	~3s	Title fades in, then a simple triangle appears with its interior angles highlighted.
2	Triangulation Concept	~6s	The triangle is split into itself (0 extra triangles). A quadrilateral appears, a diagonal is drawn, forming two triangles; interior angles of each triangle are highlighted and summed.
3	General Polygon	~8s	An $n$-gon (starting with a pentagon) is shown. A fan of diagonals from one vertex creates $n-2$ triangles. Angles of all triangles are highlighted sequentially, and the total sum is displayed as $(n-2)\times180^{\circ}$.
4	Formula Reveal	~4s	The derived formula $S = (n-2)\times180^{\circ}$ appears in large font, with a brief visual cue linking it to the number of triangles.
5	Examples Quick‑Check	~6s	Three quick examples: triangle ($n=3$), quadrilateral ($n=4$), pentagon ($n=5$). For each, the interior angle sum is shown using the formula and verified by counting highlighted angles.
6	Outro	~3s	Fade out to a concise caption: "Each added side adds one triangle → +180° to the interior sum."

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Layout

┌─────────────────────────────────────────────┐
│                TOP AREA (Title)            │
├──────────────────────┬──────────────────────┤
│                      │                      │
│   LEFT AREA          │   RIGHT AREA         │
│   (Main polygon      │   (Supporting        │
│    visual,          │    labels, angle     │
│    triangulation)   │    values)           │
│                      │                      │
├──────────────────────┴──────────────────────┤
│          BOTTOM AREA (Formula / Caption)   │
└─────────────────────────────────────────────┘

Area Descriptions

Area	Content	Notes
Top	Title: "Sum of Interior Angles of a Polygon"	Fades in during Intro, stays throughout
Left	Main visual: polygon shapes with diagonals, highlighted triangles, animated angle arcs	Primary focus; polygons grow from triangle to pentagon to generic $n$-gon
Right	Small labels showing individual angle measures (e.g., $60^{\circ}$, $90^{\circ}$) and triangle count	Appears when angles are highlighted; fades out when not needed
Bottom	Formula $S = (n-2)\times180^{\circ}$ and final caption	Appears in Phase 4 and stays for the remainder; caption appears in Outro

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Notes

• Keep total runtime under 30 seconds; durations above are approximate and can be trimmed slightly.
• Use smooth fade‑in/out and highlight transitions for angles and diagonals.
• No textual narration is required; visual cues and brief on‑screen captions convey the concept.
• The scene must be implemented as a single Manim Scene class.
• Avoid excessive text; rely on geometry and the displayed formula to communicate.