How Does the Cosine of an Angle Affect Radiation Intensity from a Black Body?

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A black body is a perfect emitter of radiation, described by the Stefan-Boltzmann law, which states it radiates energy at a rate of σT^4. The intensity of radiation is influenced by the angle between the emitting surface and the receiving area, leading to the formulation involving the cosine of that angle. This cosine factor accounts for the effective area exposed to the radiation, which varies with angle. The term "intensity" can be ambiguous, often referring to irradiance (W/m^2), which measures power per unit area. A visual representation of the areas and angles involved could enhance understanding of these concepts.
Chacabucogod
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So I understand that a black body is a diffuse emitter, and that it radiates energy at a rate of σT^4.
I also understand the correspondence between a solid angle and the superficial area of a sphere.

<br /> dS=r^2sin(θ)dθd\phi=\frac{ds}{r^2}=d\omega=\frac{dAcos(\alpha)}{r^2}<br />

What I don't understand is how they get to the formulation for intensity of radiation.

Why is the intensity of radiation proportional to the cos of the angle between the two areas? Isn't the intensity the same for every part of the half sphere since it's a black body?
 
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intensity is a vague word when it comes to radiation.
Are you talking about irradiance (W/m^2)?

A sketch of the areas and angles you're talking about might help clarify the problem
 
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