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The reason for why it is obligatory to cover the feet for women is the same as why it is obligatory for them to cover other parts of the body. In other words, the female’s body bears great attractiveness for the opposite sex, and exposing it can have negative impacts on the religious and spiritual health of the person seeing it. Nevertheless, the face and hands have been made an exception in Islam and don’t have to be covered; this might be in order to ease things up for them a bit, because covering even these parts is relatively difficult. According to Shahid Mutahhari, even these parts should have been of ones that need to be covered, but in order to make things easier for them, Allah (swt) didn’t make it obligatory for women to cover them. Of course, according to the fatwa of mujtahids, this permissibility is contingent upon them not having “decorated” (zeenah) these parts via make-up or jewelry. 
Hijab in the Quran and hadiths:
The hijab and covering is very important in Islam; its importance is to the extent that Allah (swt) says in surah Nur: “And tell the faithful women to cast down their looks and to guard their private parts, and not to display their ornaments, except for what is outward, and let them draw their scarves over their bosoms, and not display their charms except to their husbands, or their fathers, or their husband's fathers, or their sons, or their husband's sons…And let them not thump their feet to make known their hidden ornaments (like ankle bracelets)…” 
Many believe the hands and face to be exempt from the ruling this verse expresses and that there are context clues in the verse that confirm this claim:
a) When the verse makes an exception regarding visible and outward “ornaments”, regardless of whether what “ornament” refers to (let it be the actual ornament itself, or the place where the ornament has been used), it clearly shows that covering the face and hands isn't obligatory to begin with (or else the verse would have just said that they have to cover their hands and faces in the first place instead of saying that the ornaments shouldn’t be shown).
b) Another clue is that the verse says they must draw their scarves over their bosoms (which means that they have to cover the whole head, neck and chest), but says nothing about their faces.  Historical clues also suggest that covering the face with a veil wasn’t something practiced by all during the advent of Islam. 
The imams have explained this verse in many hadiths and expressed the necessary amount of covering that women need to observe. One of Imam Sadiq’s (as) companions, Fudhayl Yasar, says that he asked the imam (as): “Are the two dhira’s (from the wrist to the elbow) of women considered of the “ornaments” that Allah (swt) has said can only be shown to their spouses?” The imam (as) answered: “Yes, and anything lower than the khimar (scarf) is also considered of those ornaments…”. 
Also, Mus’adah ibn Ziyad narrates that when Imam Sadiq (as) was asked of what the ornaments that women can show are, he answered that they are the face and the two hands. 
Of course, two points must be noted here:
1- According to Islam, it is only permissible for the woman to expose her face and hands if they are plain and without any ornaments or make-up, or have been made up so little that people don’t consider it to be an ornament, and given that exposing them doesn’t entail any other problems or sin for them or others. 
2- When it is said that it is okay to expose the face and hands, it doesn’t necessarily mean that it is also permissible for men to look at them in all cases and in other words, there is no connection between the two.  
Therefore, on this basis, the viewpoint of most mujtahids is that it is wajib to cover the feet.  Of course, it isn't mandatory to cover them up to the ankles during prayer (when there aren't any non-mahrams around). 
Related Question: Question 379 (website: 390).
 With help from the Porseman software.
 Explanation: It has been said that this verse was revealed because at that time, it was customary for women to wear a scarf in a way that its end would fall on their shoulders and backs, only covering the head and the back of the neck, leaving the front of the neck and the top portion of the chest exposed. Islam fixed this and ordered that they bring the end of their scarves to the front in order to cover the chest completely. This resulted in only the face being shown.
 Tafsir Nemouneh, vol. 14, pp. 450-451.
 “ عَنِ الْفُضَیْلِ بْنِ یَسَارٍ قَالَ: سَأَلْتُ أَبَا عَبْدِ اللَّهِ ع عَنِ الذِّرَاعَیْنِ مِنَ الْمَرْأَةِ أَ هُمَا مِنَ الزِّینَةِ الَّتِی قَالَ اللَّهُ تَبَارَکَ وَ تَعَالَى وَ لا یُبْدِینَ زِینَتَهُنَّ إِلَّا لِبُعُولَتِهِنَّ قَالَ: نَعَمْ وَ مَا دُونَ الْخِمَارِ مِنَ الزِّینَةِ وَ مَا دُونَ السِّوَارَیْنِ ” Usul Kafi, pg. 5, pg. 521.
 “ عَبْدُ اللَّهِ بْنُ جَعْفَرٍ فِی قُرْبِ الْإِسْنَادِ عَنْ هَارُونَ بْنِ مُسْلِمٍ عَنْ مَسْعَدَةَ بْنِ زِیَادٍ قَالَ: سَمِعْتُ جَعْفَراً وَ سُئِلَ عَمَّا تُظْهِرُ الْمَرْأَةُ مِنْ زِینَتِهَا قَالَ: الْوَجْهَ وَ الْکَفَّیْن ” Wasailul-Shia, vol. 20, hadith 25429, pg. 203.
 See: Imam Khomeini’s istifta’at (fiqhi questions asked from a marja’), vol. 3, pg. 256, questions 33 and 34; Tawdihul-Masa’il (annotated by Imam Khomeini), vol. 2, pg. 929; Index: Women and exposing make-up, Question 379 (website: 390). More explanation on this subject and the hadiths pertaining to it have been mentioned in the marriage section of fiqhi discussions.
 For further information, see: Morteza Motahhari, Mas’eleye Hejab (The issue of Hijab), pp. 164-235/
 With help from Question 495 (website: 536).
 See: Resaleye Daneshjouyi, pg. 188, question 271. The only mujtahid who has issued a fatwa saying that it isn't necessary for women to cover their feet up to the ankles in front of non-mahrams is Ayatullah Makarem Shirazi.
 Tawdihul-Masa’el (annotated), vol. 1, pg. 442, issue 789.