General use:
* Engagement: This is the most common noun form, referring to the act of engaging, the state of being engaged, or the involvement in something.
* "They discussed the engagement of the troops."
* "The engagement ring sparkled on her finger."
Specific contexts:
* Engagement (in a battle): This refers to a battle or a period of fighting.
* "The soldiers were engaged in a fierce battle."
* "The engagement lasted for several hours."
* Engagement (of a gear): This refers to the act of meshing gears together.
* "The engine's engagement was smooth."
* Engagement (of a promise): This refers to a promise or commitment.
* "They made an engagement to meet next week."
So, the noun of "engage" is often engagement, but the specific meaning can vary depending on the context.