Area of Triangle and Circle

Description

Derives the area formulas for a triangle and a circle using visual proofs. Students see that a triangle is half a rectangle (A = ½bh), and that a circle can be rearranged into a near-rectangle whose dimensions involve π and r (A = πr²).

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Phases

#	Phase Name	Duration	Description
1	Triangle from Rectangle	10s	Draw a rectangle, then draw its diagonal to form a triangle. Shade the triangle, fade out the other half.
2	Triangle Formula	6s	Label base b and height h. Display A = ½bh.
3	Circle & Wedges	10s	Draw a circle, animate slicing into 8 sectors.
4	Rearrange to Rectangle	8s	Interleave sectors (flipping alternating ones) to form a near-rectangle.
5	Circle Formula	5s	Label width ≈ πr and height = r. Display A = πr².
6	Summary	5s	Show both formulas side by side.

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Layout

+------------------------------------------------------+
|     [Title: Area of Triangle and Circle]             |
|                                                      |
|  [Phase 1-2: Rectangle → Triangle, left side]        |
|  [Phase 3-5: Circle → wedges → near-rect, right]     |
|                                                      |
|  [Formula: A = ½bh]   [Formula: A = πr²]            |
+------------------------------------------------------+

Area Descriptions

Area	Content	Notes
Left half	Triangle derivation	Phases 1-2
Right half	Circle derivation	Phases 3-5
Bottom center	Formulas	Update per phase

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Assets & Dependencies

• Fonts: LaTeX / sans-serif
• Colors:
  • Triangle: BLUE fill
  • Rectangle outline: WHITE
  • Circle / sectors: alternating TEAL and GOLD
  • Formulas: WHITE / YELLOW
  • Background: #1a1a2e
• Manim version: ManimCE 0.19.1

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Notes

• The circle wedge rearrangement is an approximation; label it as "approximately a rectangle."
• Keep sector count manageable (8 sectors) for visual clarity.
• Both derivations happen sequentially (not simultaneously) to reduce complexity.