KVO on Swift's computed properties

That code won't work for two reasons:

1. You must add the dynamic attribute to the area property, as described in the section “Key-Value Observing” under “Adopting Cocoa Design Patterns” in Using Swift with Cocoa and Objective-C.

2. You must declare that area depends on width and height as described in “Registering Dependent Keys” in the Key-Value Observing Programming Guide. (This applies to Objective-C and Swift.) And for this to work, you also have to add dynamic to width and height.

   (You could instead call willChangeValueForKey and didChangeValueForKey whenever width or height changes, but it's usually easier to just implement keyPathsForValuesAffectingArea.)

Thus:

import Foundation

class MyObject: NSObject {

    @objc dynamic var width: Double = 0
    @objc dynamic var height: Double = 0

    @objc dynamic private var area: Double {
        return width * height
    }

    @objc class func keyPathsForValuesAffectingArea() -> Set<String> {
        return [ "width", "height" ]
    }

    func register() {
        self.addObserver(self, forKeyPath: "area", options: [ .old, .new ], context: nil)
    }

    override func observeValue(forKeyPath keyPath: String?, of object: Any?, change: [NSKeyValueChangeKey : Any]?, context: UnsafeMutableRawPointer?) {
        print("observed \(keyPath) \(change)")
    }
}

let object = MyObject()
object.register()
object.width = 20
object.height = 5

Output:

observed Optional("area") Optional([__C.NSKeyValueChangeKey(_rawValue: new): 0, __C.NSKeyValueChangeKey(_rawValue: kind): 1, __C.NSKeyValueChangeKey(_rawValue: old): 0])
observed Optional("area") Optional([__C.NSKeyValueChangeKey(_rawValue: new): 100, __C.NSKeyValueChangeKey(_rawValue: kind): 1, __C.NSKeyValueChangeKey(_rawValue: old): 0])

As @Rob stated in his answer, make area dynamic to be observed from objective-c

Now add willSet { } and didSet { } for width and height properties,
inside willSet for both properties add this self.willChangeValueForKey("area") and in didSet add self.didChangeValueForKey("area");

Now observers of area will be notified every time width or height change.

Note: this code is not tested, but I think it should do what expected