It's greatest extent was in about 500 BCE. It stretched from Libya-Egypt through the Middle East to Central Asia and today's Pakistan.
Those who established the Persian Empire to its greatest extent were Cyrus the Great, Cambyses II and Darius the Great.
During Darius the Great's reign at the end of the 6th Century BCE.
It had stretched from Libya in the west to modern Pakistan in the east. However by 330 BCE it was disintegrating and was largely in the control of the Macedonians led by Alexander the Great who was converting it into an empire of his own.
A Persian Empire existed in various forms from ancient times until the early twentieth century when it was renamed the Empire of Iran. The Empire of Iran was replaced with the Islamic Republic of Iran in 1979. At it's greatest extent the Persian Empire under Darius the Great ruled all of modern Iran, Afghanistan, Iraq, Syria and Turkey.
Cyrene in Libya.
I guess you refer to Darius I (the Great) the third king of the Achaeminid Empire (the Persian Empire at its greatest extent). He divided the empire into provinces and placed satraps (governors) to administer them. It was a creation of administrative subdivisions.
The Roman Empire reached its greatest extent under Trajan in 116 AD.
The Roman Empire reached its greatest extent in 116 AD under the emperor Trajan.
From Libya to Central Asia.
The Roman empire reached its greatest extent under the reign of Trajan.
Alexander conquered Egypt, Macedonia, and the Persian Empire.
From Libya to Central Europe - the same as the Persian Empire from whom he conquered/stole it.