Directivity is the ratio of the area of a sphere (4πr2) to the beam area A of the antenna

$$\boxed {D = \frac{4π}{\iint_{4π}P_n(θ, ϕ)dΩ}= \frac{4π}{Ω_A} \ \ \ \ \ \ \ \ \ Directivity \ from \ beam \ area \ Ω_A}$$

$Where \ P_n(θ, ϕ) dΩ = P(θ, ϕ)/ P(θ, ϕ)_{max} = normalized \ power \ pattern$

The smaller is the beam area, the larger the directivity D. For an antenna that radiates over only half a sphere the beam area $A = 2πr^2$ (Fig. given) and the directivity is

D = 4π/2π = 2 (= 3.01 dBi) where dBi = decibels over isotropic

Note that the idealized isotropic antenna radiated over complete sphere $(A = 4πr^2)$ hence lowest possible directivity for it is

$D = 4π/4π =1$.

All actual antennas have directivities greater than 1 (D > 1). The simple short dipole has a beam area $A = 2.67πr^2$ and a directivity D = 1.5 (= 1.76 dBi).