The following query can be used to detemine tablespace and other params:
select df.tablespace_name "Tablespace",
totalusedspace "Used MB",
(df.totalspace - tu.totalusedspace) "Free MB",
df.totalspace "Total MB",
round(100 * ( (df.totalspace - tu.totalusedspace)/ df.totalspace)) "Pct. Free"
from (select tablespace_name,
round(sum(bytes) / 1048576) TotalSpace
from dba_data_files
group by tablespace_name) df,
(select round(sum(bytes)/(1024*1024)) totalusedspace,
tablespace_name
from dba_segments
group by tablespace_name) tu
where df.tablespace_name = tu.tablespace_name
and df.totalspace <> 0;
Source: https://community.oracle.com/message/1832920
For your case if you want to know the partition name and it's size just run this query:
select owner,
segment_name,
partition_name,
segment_type,
bytes / 1024/1024 "MB"
from dba_segments
where owner = <owner_name>;