how to understand “the value of the ratio”？
"Can you tell me what these objects have in common?" "Well, they are the same height and the mouth seems the same size."
"The value of the ratio of the semi-sphere is equal to the height of the cone," corrected Archimedes.

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I think there is not enough information. a ratio compares two quantities, and I don't see which two quantities are being compared

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"value"  = the measurement, or "numerical value"

The height of the cone is a "value" also,  a   "numerical value"