Decays of radionuclides throughout the Earths interior produce geothermal heat, but also are a source of antineutrinos. The (angle-integrated) geoneutrino flux places an integral constraint on the terrestrial radionuclide distribution. In this paper, we calculate the angular distribution of geoneutrinos, which opens a window on the differential radionuclide distribution. We develop the general formalism for the neutrino angular distribution, and we present the inverse transformation which recovers the terrestrial radioisotope distribution given a measurement of the neutrino angular distribution. Thus, geoneutrinos not only allow a means to image the Earths interior, but offering a direct measure of the radioactive Earth, both (1) revealing the Earths inner structure as probed by radionuclides, and (2) allowing for a complete determination of the radioactive heat generation as a function of radius. We present the geoneutrino angular distribution for the favored Earth model which has been used to calculate geoneutrino flux. In this model the neutrino generation is dominated by decays in the Earths mantle and crust; this leads to a very ``peripheral angular distribution, in which 2/3 of the neutrinos come from angles > 60 degrees away from the downward vertical. We note the possibility of that the Earths core contains potassium; different geophysical predictions lead to strongly varying, and hence distinguishable, central intensities (< 30 degrees from the downward vertical). Other uncertainties in the models, and prospects for observation of the geoneutrino angular distribution, are briefly discussed. We conclude by urging the development and construction of antineutrino experiments with angular sensitivity. (Abstract abridged.)
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