"The sons of Israel served Eglon the king of Moab eighteen years. But when the sons of Israel cried to the LORD, the LORD raised up a deliverer for them, Ehud the son of Gera, the Benjamite, a left-handed man. And the sons of Israel sent tribute by him to Eglon the king of Moab. Ehud made himself a sword which had two edges, a cubit in length, and he bound it on his right thigh under his cloak. He presented the tribute to Eglon king of Moab. Now Eglon was a very fat man. It came about when he had finished presenting the tribute, that he sent away the people who had carried the tribute. But he himself turned back from the idols which were at Gilgal, and said, “I have a secret message for you, O king.” And he said, “Keep silence.” And all who attended him left him. Ehud came to him while he was sitting alone in his cool roof chamber. And Ehud said, “I have a message from God for you.” And he arose from his seat. Ehud stretched out his left hand, took the sword from his right thigh and thrust it into his belly. The handle also went in after the blade, and the fat closed over the blade, for he did not draw the sword out of his belly; and the refuse came out." -Judges 13:14-22
In regards to the secret meeting, is it possible that Eglon having conquered the peoples who are giving tribute (Jdg 3:14) would have desired to hear of any information and might have suspected that Ehud was going to tell him of sedition? or another possibility is that he thought Ehud may have been seeking to gain favour. Either way the secret meeting would have an array of potential benefits or upsides. Eglon, thinking Ehud was unarmed (Jdg 3:16) felt completely safe.
Being left handed let him hide the sword on the opposite side as to where it would normally be. I suppose when the Eglon’s guards checked to see if he had a weapon Ehud must have shown them the traditional side and there would have been nothing there. Given that Ehud was also giving tribute for either his people or city/town/area (Jdg 3:17) after having being conquered for 18 years (Jdg 3:14), there would be little reason as to why this tribute would be any different than the last. Thus being left handed he was able to sneak a weapon into the secret meeting in Judges 3:19 and then also surprise him without any fighting back in Judges 3:21-22. The defencelessness of Eglon also may have come about through thinking Ehud was unarmed (Jdg 3:16), thus leaving Eglon to feel completely safe.
"A cubit in length":
"The Hebrew word 'gomed' is used, in this verse, to apply to a double-edged dagger which is about a foot long. The NLT suggests it is about 12 inches long, whereas the NIV suggests 18 inches. The point is (sorry, no pun intended) this dagger could be strapped to a man's thigh. A dagger between 12 and 18 inches in length might work. Certainly, nothing longer. (Source: New Living Translation Study Bible notes (page 424).")
A hefty dagger. Given the size I wonder what weight Eglon was (Jdg 3:22). The height to weight ratio would also have an impact.