Short answer
Only Allah knows: "none knows the soldiers of your Lord except Him" (74:31). The hadith show how vast their number is: seventy thousand angels enter al-Bayt al-Ma'mur in heaven every day and never return to it, and there is no space in the heavens without an angel in worship.
Belief in the angels in general is fard, without needing to know their number. Belief in the angels named in the Quran and authentic hadith is also required.
The four schools agree on this.
Explanation
The Quran mentions that Hell has nineteen keepers (74:30), and then explains that the number is a test, adding: "And none knows the soldiers of your Lord except Him" (74:31). The angels are among those countless soldiers.
On the Night Journey the Prophet ﷺ was shown al-Bayt al-Ma'mur, the "frequented house" in the seventh heaven. Jibril told him that seventy thousand angels pray in it every day, and when they leave they never return to it (Sahih al-Bukhari). Imagine that number every day since creation.
The Prophet ﷺ also said the heaven creaks, and has the right to creak, because there is no space in it the size of four fingers without an angel placing his forehead in prostration to Allah (Tirmidhi, graded hasan). And on the Day of Judgement Hell will be brought with seventy thousand reins, each held by seventy thousand angels (Sahih Muslim).
Besides this, every person has angels assigned to them: recorders, guardians and others. Only a handful are named: Jibril, Mikail, Israfil, Malik, Harut and Marut, and Munkar and Nakir. See the four great angels, guardian angels and the seven heavens.
The evidence
Quran, Surah Al-Muddaththir 74:31
وَمَا جَعَلْنَآ أَصْحَٰبَ ٱلنَّارِ إِلَّا مَلَٰٓئِكَةًۭ ۙ وَمَا جَعَلْنَا عِدَّتَهُمْ إِلَّا فِتْنَةًۭ لِّلَّذِينَ كَفَرُوا۟ لِيَسْتَيْقِنَ ٱلَّذِينَ أُوتُوا۟ ٱلْكِتَٰبَ وَيَزْدَادَ ٱلَّذِينَ ءَامَنُوٓا۟ إِيمَٰنًۭا ۙ وَلَا يَرْتَابَ ٱلَّذِينَ أُوتُوا۟ ٱلْكِتَٰبَ وَٱلْمُؤْمِنُونَ ۙ وَلِيَقُولَ ٱلَّذِينَ فِى قُلُوبِهِم مَّرَضٌۭ وَٱلْكَٰفِرُونَ مَاذَآ أَرَادَ ٱللَّهُ بِهَٰذَا مَثَلًۭا ۚ كَذَٰلِكَ يُضِلُّ ٱللَّهُ مَن يَشَآءُ وَيَهْدِى مَن يَشَآءُ ۚ وَمَا يَعْلَمُ جُنُودَ رَبِّكَ إِلَّا هُوَ ۚ وَمَا هِىَ إِلَّا ذِكْرَىٰ لِلْبَشَرِ ﴿٣١﴾
And We have not made the keepers of the Fire except angels. And We have not made their number except as a trial for those who disbelieve - that those who were given the Scripture will be convinced and those who have believed will increase in faith and those who were given the Scripture and the believers will not doubt and that those in whose hearts is hypocrisy and the disbelievers will say, "What does Allah intend by this as an example?" Thus does Allah leave astray whom He wills and guides whom He wills. And none knows the soldiers of your Lord except Him. And mention of the Fire is not but a reminder to humanity.
The verse that ends by saying none knows the soldiers of your Lord but Him.
Sahih al-Bukhari 3207
Sahih... يصلي فيه كل يوم سبعون ألف ملك ...
"Seventy thousand angels pray in it every day."
Shows the vast number of angels.
Jami' at-Tirmidhi 2312
Grading to be checked... أطت السماء وحق لها أن تئط ...
"The heaven creaks, and it has the right to creak."
The heavens are full of angels in worship.
Sahih Muslim 2842
Sahih... يؤتى بجهنم يومئذ لها سبعون ألف زمام ...
"Hell will be brought that Day with seventy thousand reins."
Each rein is held by seventy thousand angels.
Where the ruling is recorded
- al-Fiqh al-Akbar (Imam Abu Hanifa), Opening section on the articles of belief. Lists belief in the angels among the foundations of iman.
- Tafsir Ibn Kathir, Commentary on Surah al-Muddaththir, verse 31. Discusses the number of angels and the reports about al-Bayt al-Ma'mur.
Checked by our team. For a more accurate answer to your own situation, consult a scholar. Quran translation by Sahih International. Hadith Arabic from the standard editions of the six books and the Muwatta, with gradings as recorded there; the English is our own rendering. Spotted a mistake? Tell us.