How do you measure the area and display it in the views

Can the area measurement display be displayed in the views as well as the Angle display?like this.

Yes, in recent Slicer Preview Releases you can use closed curve markups to measure area and display in views (you can select list of enabled measurements in Measurements section in Markups module).

Can i use the plane makeups to measure area?And then how can i implement it in code, or which interface i can use?

In latest Slicer Preview Releases, you can enable “area” measurement for markups plane nodes (in Markups module / Measurements section).