Lumen (lm) is the SI derived unit of luminous flux, which measures the total amount of light emitted by a source in all directions. Candela (cd) is the SI base unit of luminous intensity, which measures the amount of light emitted by a source in a particular direction.

The formula for converting lumen to candela is:

\[I = \frac{\Phi}{2\pi (1 - \cos(\frac{\theta}{2}))}\]Where:

- \(I\) = Luminous intensity (candela)
- \(\Phi\) = Luminous flux (lumens)
- \(\theta\) = Apex angle (radians)

- Identify the luminous flux in lumens (\(\Phi\)) and the apex angle in degrees (\(\theta\)).
- Convert the apex angle from degrees to radians by multiplying by \(\frac{\pi}{180°}\).
- Apply the formula: \(I = \frac{\Phi}{2\pi (1 - \cos(\frac{\theta}{2}))}\).
- The result is the luminous intensity in candela (\(I\)).

Let's calculate the luminous intensity for a light source with 1000 lumens and an apex angle of 60°:

- \(\Phi = 1000 \text{ lm}\), \(\theta = 60°\)
- \(\theta \text{ in radians} = 60° \times \frac{\pi}{180°} = \frac{\pi}{3} \text{ radians}\)
- \[\begin{align} I &= \frac{1000}{2\pi (1 - \cos(\frac{\pi}{6}))} \\ &\approx \frac{1000}{2\pi \times 0.1340} \\ &\approx 1189.2 \text{ candela} \end{align}\]

This diagram illustrates how a 1000 lumen light source with an apex angle of 60° produces a luminous intensity of approximately 1189.2 candela. The red point represents the light source, and the yellow cone shows the spread of light based on the apex angle.

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