What do you call those high areas that give you a full view of the city?

Solution 1:

A common term for a place that affords such a view would be a scenic overlook or simply a scenic view:

A scenic overlook, or just an overlook, observation point, lookout, viewing point or vista point is a high place where people can gather to view scenery (often with binoculars), and to photograph it. Scenic overlooks are typically created alongside mountain roads, often as a simple turnouts where motorists can pull over onto pavement, gravel, or grass on the right-of-way. Many are larger, having parking areas, while some (typically on larger highways) are off the road completely.

Here are some signs to such:

An older word, now more restricted to a piece of architecture meant for such purposes, is a belvidere or belvedere.

Solution 2:

It is called a viewpoint. (single word for viewing point)

Sometimes called vantage point as well.

viewpoint: a place from which something can be viewed

All the synonyms are mentioned in this article also:

http://en.wikipedia.org/wiki/Overlook

"Overlook" and "scenic overlook" is usually used for natural and mountain views. But to be more specific, you can always say "city overlook" or "city viewpoint".

Additions from Andrew Leach:

There is even a viewpoint sign:

_{Source: http://www.followthebrownsigns.com/viewpoint/}

Viewpoint road sign:

_{Source: http://latterdaymusings.blogspot.co.uk/2013/06/road-trip-oregon-jewel-mist-fishawk.html}

It is also called a scenic viewpoint. It is used in USA too. When you say scenic view, it is the view that you are going to see but the viewpoint is the point where you see that view from.

_{Source: http://portlandwomensforum.com/photos.html}

_{Source: http://www.milespointsandmaitais.com/category/uncategorized/}

Solution 3:

A Lookout.

"On that hill over there there is a lookout overseeing the whole city."

"Mary said she would come with me to point lookout this saturday."