It’s tempting to think that dissect and bisect mean the same thing : to cut something in two. After all, the bi- and di- prefixes both convey the meaning of “two” or “twice”. So we have bicycle (with two wheels) and dioxide (with two oxygen atoms).
In fact only bisect means to cut in two. A line cutting through a circle would be said to bisect it. A river might bisect a city. The prefix of dissect is not di-, however, but dis- meaning “apart”. It’s the same prefix as in words such as disintegrate (to break up into small parts) and dismember (to cut the limbs from something).
Dissect means to methodically cut something into pieces rather than to merely cut it in two. So, for example, organs might be dissected in order to find out how they work. Dissect has also acquired the more figurative meaning of studying something in great detail without any literal cutting taking place. Thus a statement or report might be “dissected” to see if it contains any inconsistencies or inaccuracies.
When the two words are mixed up, generally it’s the case that dissect is used when bisect is meant. Sometimes dissect is misspelled “disect” to add to the confusion. Only bisect means to cut something into two parts.