These six words are all pronounced はかる, and many are listed under the same definitions in dictionaries.
What are the difference between these words?
計る - to measure (quantities and in general)
計る is used for counting/measuring the number of something. It can also mean "to plan". (General word for measuring things)
量る - to measure (a three dimensional amount)
量る is used to measure weight and volume.
測る - to measure (lengths and areas)
測る is used to measure length, height, etc.
図る - to plan
図る is used to plan or aim towards things.
謀る - to plot
謀る is used to plot or aim towards bad things. (Same as 図る but for bad things)
諮る - to discuss
諮る is used to indicate the exchange of opinions.
Information comes from the disambiguation notes in 大辞林 for はかる.
General meaning: To plan out; to measure
measure, take; weigh; sound; time; gauge
Use 計る when examining the time or extent of something
consult (with); submit
諮る means “to get the opinion of another person” or “to consult for advice”.
plan; conspire, plot; try, attempt; seek; aim for; work for
図る means “to form a plan, with the intent to make it real.” If it’s something you’re setting a goal to do [e.g. 目指す], then use 図る
測る means “to measure length and/or breadth.”
謀る means “to deceive”.
Similar to 図る above, 謀る also means to consider from a variety of angles, take measurements, and execute a plan so that an existing objective can become a reality.
If you’re planning something evil, 謀る is commonly used for this purpose
量る means “to measure weight or capacity” or “to estimate”.