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.　©　2024