Tragic royal's damaged pectoral area (9)

I believe the answer is:

cleopatra

I'm a little stuck... Click here to teach me more about this clue!

'tragic royal's' is the definition.
'Cleopatra' can be an answer for 'royal's' (Cleopatra is an example). I am not certain of the 'tragic' bit.

'damaged pectoral area' is the wordplay.
'damaged' is an anagram indicator.
'area' becomes 'a' (maths abbreviation).
'pectoral' with letters rearranged gives 'cleopatr'.
'cleopatr'+'a'='CLEOPATRA'

Can you help me to learn more?

(Other definitions for cleopatra that I've seen before include "Good part for actress" , "tragic role" , "Film" , "Old queen of Egypt" , "Queen of Egypt, loved by Caesar and Mark Antony" .)

I've seen this clue in The Telegraph.

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