The words sighed, side sound the same but have different meanings and spellings. Why do sighed, side sound the same even though they are completely different words?
The answer is simple: sighed, side are homophones of the English language.
Mathematics A line bounding a plane figure.
Mathematics A surface bounding a solid figure.
A surface of an object, especially a surface joining a top and bottom: the four sides of a box.
A surface of an object that extends more or less perpendicularly from an observer standing in front: the side of the ship.
Simple past tense and past participle of sigh.
Definitions from The American Heritage® Dictionary of the English Language, 4th Edition, from Wiktionary, Creative Commons Attribution/Share-Alike License and Wordnik.
Homophones (literally "same sound") are usually defined as words that share the same pronunciation, regardless of how they are spelled.