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

The formula for converting candela to lumen is:

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

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

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

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

- \(I = 1000 \text{ cd}\), \(\theta = 60°\)
- \(\theta \text{ in radians} = 60° \times \frac{\pi}{180°} = \frac{\pi}{3} \text{ radians}\)
- \[\begin{align} \Phi &= 2\pi \times 1000 \times (1 - \cos(\frac{\pi}{6})) \\ &\approx 2\pi \times 1000 \times 0.1340 \\ &\approx 841.4 \text{ lumens} \end{align}\]

This diagram illustrates how a 1000 candela light source with an apex angle of 60° produces a luminous flux of approximately 841.4 lumens. 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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