A cooling system for a turbomachine blade including a multi-wall blade and a platform. A cooling circuit for the multi-wall blade includes: a central cavity and a plurality of outer cavities, wherein the central cavity comprises an intermediate passage of the cooling circuit, and wherein a flow of cooling air is fed into the cooling circuit through a first outer cavity of the plurality of outer cavities; and a connection for fluidly connecting a second outer cavity of the plurality of outer cavities to a platform core of the platform, the flow of cooling air passing through the connection into the platform core of the platform.