This　article　focuses　on　the　issue　of　reachable　set　estimation(RSE)　with　the　consensus　performance　for　multi-agent　systems(MASs)　under　event-triggered　scheme(ETS)　and　semi-Markov　switching　topologies　with　partially　unknown　rates(SMST-PUR).　Firstly,　through　model　transformation,　the　consensus　research　of　MASs　can　be　converted　to　the　closed-loop　stability　analysis　problem,　and　a　periodic　sampling　ETS　is　employed　to　reduce　the　frequency　of　information　broadcasting　by　the　behavior　of　communicating　with　neighbors　at　the　special　moments.　Secondly,　the　Lyapunov-Krasovskii　functions(LKFs)　with　state　information　of　time　delay　are　constructed,　whose　goal　is　to　capture　more　dynamics　of　time　delay　that　impact　on　the　controller＇s　design.　And　a　group　of　sufficient　conditions　in　the　aspect　of　linear　matrix　inequalities(LMIs)　is　derived　to　guarantee　that　the　states　of　error　system　can　be　confined　within　a　finite　union　of　ellipsoid-like　sets　corresponding　to　each　mode　with　consensus　manifestation　under　the　action　of　switching　topology　randomly　governed　by　SMST-PUR.　Simultaneously,　the　method　of　co-designing　the　consensus　controller　and　the　triggered　matrices　of　ETS　with　RSE　performance　is　depicted　to　save　communication　resources　availably.　Finally,　a　study　case　is　utilized　to　verify　the　validity　of　the　proposed　method　and　the　comparative　results　illustrate　the　advantages　and　effectiveness　of　the　control　algorithm　designed　in　this　paper.