# How is antenna gain correlated to beam width?

Let's say you have two dipole type antennas. Antenna A has a gain of 2.15 dBi, a horizontal beam width of 360 deg and a vertical beam width of 45 deg. Antenna B is similar to antenna A, but has a horizontal beam width of 360 deg and a vertical beam width of 42 deg. Can you use the ratio of the vertical beam widths to predict the gain of antenna B?

Note: In the application I'm asking about I'm not sure what method they used to calculate the beam widths. Maybe someone else knows which methods are most commonly used for dipole antennas.

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Antenna gain is often expressed in the following form,

$G = \frac{4\pi A_{e}}{\lambda^{2}}$,

where $A_{e}$ is the effective area of the antenna and $\lambda$ is the operating wavelength. However, using the antenna equation, the effective area can be expressed in terms of the main beam width (3dB width) $\Omega$,

$A_{e} = \frac{\lambda^{2}}{\Omega}$.

Assuming both antennas operate at the same wavelength, the following is true,

$G_{B} = G_{A}\frac{\Omega_{B}}{\Omega_{A}}$.