Hi
I don't think 10 foot will make much of a difference to an 80 foot tree.
You could cut your eucalyptus tree to just above the ground if you want and it will regrow (as a multi stem tree or bush). This is called coppicing. Altenatively if you cut it back to bare stem about 3 foot above the ground it will also regrow ... pollarding. This really should be done in the early spring, but if it is annoying you, I would have a go at it now. Have faith .... eucalyptus usually grow about 3 - 4 feet a year. The new growth after cutting back is lovely.