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Arcs & Sectors

It's all about ratios. Part over Whole.

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Geometry

Radian-Arc Conversion

\[ s = r\theta \]

If the angle \(\theta\) is in radians, Arc Length \(s\) is just radius times angle. Easier than degrees.

📝 Example: Radian Arc

Radius 10, Angle \(3\) radians. Find arc length.

\(s = 10 \cdot 3 = 30\)

Result: 30.

Geometry

Arc Length Formula

\[ L = \frac{\theta}{360} \times 2\pi r \]

Think: "Fraction of the Circumference".

📝 Example: Pizza Crust

Radius 8, Angle \(45^\circ\). Find arc length.

\(L = 2\pi(8) \cdot \frac{45}{360} = 16\pi \cdot \frac{1}{8} = 2\pi\)

Result: 2π.

Geometry

Sector Area Formula

\[ A = \frac{\theta}{360} \times \pi r^2 \]

Think: "Fraction of the Area".

📝 Example: Pizza Slice

Radius 10, Angle \(72^\circ\). Find sector area.

\(A = \pi(10)^2 \cdot \frac{72}{360} = 100\pi \cdot \frac{1}{5} = 20\pi\)

Result: 20π.