• 遊園地ゆうえんち — amusement park
• 遊学ゆうがく — studying abroad
• 遊歩道ゆうほどう — promenade, walkway
• 外遊がいゆう — overseas trip (esp. by a diplomat)

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遊 is a phono-semantic compound: the part with 斿 gives the sound, and 辶 (movement radical) gives the meaning of "wandering or moving." Together, they reflect the idea of playful movement or free activity.

子供達こどもたちが公園こうえんで遊あそんでいます。
The children are playing in the park.

夏休なつやすみに遊園地ゆうえんちへ行いきました。
I went to an amusement park during summer vacation.