This is my first post here, so I hope this helps.

Depends on how exactly you want to measure it:

1. In terms of landmass it peaked in the early 1920s at 22%. This was after the scramble for Africa and the World War 1 peace treaties which established numerous new British mandates.

2. In terms of naval power, the British essentially dominated throughout the 19th Century, in a scale only beaten by the US today. It was only the Anglo-German arms race at the turn of the 20th century which started applying a limited amount of pressure.

3. In terms of geopolitical power, Britain in the 1870s and 1880s was very strong, but far from being free of rivalries. The Scramble for Africa would see continued friction with the French over interests, particularly in Egypt. It was not the case Britain could walk unchallenged into territories they wanted.

4. In terms of industrial output (coal, steel etc), the British still held a commanding lead in the 1870s and 1880s, but already by this point it was being steadily eroded by new opponents such as newly reunified Germany. I haven't got the figures here to hand, but I believe the British advantage peaked in middle of the 19th century.

5. In terms of the security of the empire amongst those it ruled, it was not until the First World War that nationalism began to become a serious force in colonies, which would ultimately be the driving force towards decolonisation.

There are lots of other measures, but in short I don't think you can conclusively say the 1870s and 1880s were the height of the Empire. Historians generally refer to the end of the Napoleonic Wars up until the First World War as being Pax Britannica, a period where the British Empire was the global power. I would say this entire period would be the height of the Empire. It would exist before and after but never as the global power.