GeoTools is a Java library which can read and write GeoJSON and provides access to features like area via the JTS library. It will also help you to reproject the geometries from the lat/lon of GeoJSON to a suitable projection for calculating areas in m^2.
You can do this several ways, I think that the easiest (if you only have a few GIS things to do) is to use Turf see here There are also a couple of GIS library whom are a bit heavier (size wise) leaflet: how to After you can also try openlayers3
The map displays most likely perfectly fine, but you provide the wrong coordinates. The bing map, if I'm not mistaken, uses the web mercartor projection (EPSG:3857), yet you seem to provide geographic coodinates. Try to convert them using var styles = [ 'Road', 'Aerial', 'AerialWithLabels', 'collinsBart', 'ordnanceSurvey' ]; var layers = ...
imagerySet is a specific attribute for bing maps. It wont work in any other tile type. getImagerySet wont work as there is no such function documented on ol3. Also seems that ol3 have no function to retrieve the bing image type. You can always pass a custom attribute during layer initilasation and the retrive it using the get method. Check the following ...
I can get your point but I really can not understand what do you expect to visually get out of it. Would it be somehtig like this example? -->clipping. Show the a map but hide the areas you need to be not visisble. Or would it be something like this example.-->extent+minZoom+maxZoom prevent the user moving the map in areas you need to be hidden and thus ...
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