Branches of external carotid artery in carotid triangle includes:

1) Superior thyroid artery:

Arises below the level of the greater cornu of hyoid dividing into terminal branches at the apex of the thyroid lobe.

Branches: Infrahyoid, Sternomastoid and Superior laryngeal branches.

2) Lingual artery: 

Arises behind the lip of the greater cornu of the hyoid bone. It passes deep and medial to the hyoglossus. Supplies tongue, buccal mucosa and floor of mouth.

Branches: Suprahyoid, dorsal lingual, sublingual artery.

3) Facial artery (External maxillary artery):

Arises anteriorly above the lingual artery. This artery passes deep to the posterior belly of the digastric. Pulsations of this artery are palpable at the base of the mandible. It supplies the face, submandibular gland and the tonsil.

Branches: Ascending palatine, tonsillar, submental, inferior labial, superior labial, lateral nasal.

4) Ascending pharyngeal artery:

This is the smallest branch of the external carotid artery. Arises posteriorly near the origin of the external carotid artery and ascends between the internal carotid artery and pharynx.

Branches: Pharyngeal, inferior tympanic and meningeal branches.

5) Occipital artery:

Arises posteriorly along posterior belly of digastric crossing the internal carotid artery, internal jugular vein, hypoglossal, vagus and accessory nerves.

Branches: Sternomastoid, mastoid, auricular, meningeal, occipital, descending branches.

6) Posterior Auricular artery:

Arises from the external carotid artery, above the digastric muscle and stylohyoid muscle, opposite the apex of the styloid process.

Branch: Stylomastoid branch.

Branches outside the carotid triangle:

1) Superficial temporal artery: 

Arises at the neck of the mandible. It supplies the parotid gland, temporomandibular joint and masseter.

Branches: Transverse facial artery, auricular branches, middle and posterior temporal artery.

2) Internal maxillary artery:

It is the terminal branch of the external carotid artery. Supplies the mandible, maxilla, teeth, palate, muscles of mastication and cranial dura mater.

Branches: Divided into mandibular part, pterygoid part and pterygopalatine part. Each part gives five branches.

Surgical importance:

1. The external carotid artery can be approached by an incision made along the anterior border of sternomastoid first above the angle of mandible and downwards. It can be differentiated from the internal carotid because it gives branches in the neck unlike the internal and common carotid.
2. At the bifurcation of the common carotid, the internal carotid is lateral to external carotid.
3. Ligation of the external carotid artery is done in severe tonsillar bleeding after it gives the first branch, the superior thyroid artery. This is done because a thrombus is formed distal to the ligation site if dislodged, would enter the superior thyroid artery and not the internal carotid artery. Also, the first branch of the external carotid artery helps to differentiate it from the internal carotid artery which has no branches in the neck.