Sum of angles of a triangle

In a <a href="/facts/Euclidean_space/R2UbzmzM">Euclidean space</a>, the <a href="/facts/Summation/V2WUxoKn">sum</a> of angles of a triangle equals a <a href="/facts/Straight_angle/gFUTEQYq">straight angle</a> (180 <a href="/facts/Degree_(angle)/f48Ns8SX">degrees</a>, <a href="/facts/Pi/vC0UBj1q">π</a> <a href="/facts/Radians/Srd38Sxa">radians</a>, two <a href="/facts/Right_angle/psaslRq7">right angles</a>, or a half-<a href="/facts/Turn_(geometry)/QczxAyuM">turn</a>). A <a href="/facts/Triangle/cRXUwsMn">triangle</a> has three angles, one at each <a href="/facts/Vertex_(geometry)/ID3cea9z">vertex</a>, bounded by a pair of adjacent <a href="/facts/Edge_(geometry)/3lFN6sEz">sides</a>.
The sum can be computed directly using the definition of angle based on the <a href="/facts/Dot_product/tNz8MLjT">dot product</a> and <a href="/facts/Trigonometric_identities/Xw9Jx8Vz">trigonometric identities</a>, or more quickly by reducing to the two-dimensional case and using <a href="/facts/Euler%27s_identity/QB4HQun8">Euler's identity</a>.
It was unknown for a long time whether other <a href="/facts/Geometry/wTQf5ffQ">geometries</a> exist, for which this sum is different. The influence of this problem on mathematics was particularly strong during the 19th century. Ultimately, the answer was proven to be positive: in other spaces (geometries) this sum can be greater or lesser, but it then must depend on the triangle. Its difference from 180° is a case of <a href="/facts/Angular_defect/FMHKeUVS">angular defect</a> and serves as an important distinction for geometric systems.

Sum of angles of a triangle open-in-new

Sum of angles of a triangle