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The beam solid angle of an antenna is the integral of its normalized radiated-power pattern over the complete sphere. If $U(\theta ,\varphi )$ is radiated power per unit solid angle and $U_{\max }$ is its maximum, thenIt is measured in steradians and includes sidelobes and backlobes, not just a chosen main-beam boundary. Since total radiated power is $P_{\mathrm{rad}}=\int Ud\Omega$, peak antenna directivity satisfiesThe definition relates concentration of radiation to the entire pattern, rather than to a single width measurement.^[ref]

For a narrow, axisymmetric Gaussian power pattern $U/U_{\max }=e^{-{\theta }^2/(2{\sigma }^2)}$, replace $\sin \theta$ by $\theta$ near the axis and extend the negligible tail to infinity. The resulting Gaussian beam solid angle is ${\Omega }_A\simeq 2\pi {\sigma }^2$. Its full half-power width is ${\theta }_{\mathrm{HP}}=2\sqrt{2\ln 2}\sigma$, so ${\Omega }_A\simeq \pi {\theta }_{\mathrm{HP}}^2/(4\ln 2)$. These approximations require a narrow beam with no significant additional lobes; a broad pattern or strong sidelobes requires the full integral.