how to find angles of a triangle

Triangle Figure
Angle-Side-Angle (ASA)

Triangle Diagram for Angle-Side-Angle with Angles A, B and C and sides opposite those angles a, b and c respectively
A = angle A
B = angle B
C = angle C
a = side a
b = side b
c = side c

P = perimeter
s = semi-perimeter
K = area
r = radius of inscribed circle
R = radius of circumscribed circle

Calculator Use

Each calculation option, shown below, has sub-bullets that list the sequence of methods used in this calculator to solve for unknown angle and side values including Sum of Angles in a Triangle, Law of Sines and Law of Cosines. These are NOT the ONLY sequences you could use to solve these types of problems.

See also these Trigonometry Calculators:
Law of Cosines Calculator
Law of Sines Calculator

Solving Triangle Theorems

Base Triangle ABC

AAA is Angle, Angle, Angle

Specifying the three angles of a triangle does not uniquely identify one triangle. Therefore, specifying two angles of a tringle allows you to calculate the third angle only.

Given the sizes of 2 angles of a triangle you can calculate the size of the third angle. The total will equal 180° or π radians.

C = 180° - A - B (in degrees)

C = π - A - B (in radians)

AAS is Angle, Angle, Side

Given the size of 2 angles and 1 side opposite one of the given angles, you can calculate the sizes of the remaining 1 angle and 2 sides.

use the Sum of Angles Rule to find the other angle, then

use The Law of Sines to solve for each of the other two sides.

ASA is Angle, Side, Angle

Given the size of 2 angles and the size of the side that is in between those 2 angles you can calculate the sizes of the remaining 1 angle and 2 sides.

use the Sum of Angles Rule to find the other angle, then

use The Law of Sines to solve for each of the other two sides.

ASS (or SSA) is Angle, Side, Side

Given the size of 2 sides (a and c where a < c) and the size of the angle A that is not in between those 2 sides you might be able to calculate the sizes of the remaining 1 side and 2 angles, depending on the following conditions.

For A ≥ 90° (A ≥ π/2):

If a ≤ c there there are no possible triangles

Example:
ASS Theorem A > 90 and side a less than side c

If a > c there is 1 possible solution

use The Law of Sines to solve for angle C
use the Sum of Angles Rule to find the other angle, B
use The Law of Sines to solve for the last side, b
Example:

For A < 90° (A < π/2):

If a ≥ c there is 1 possible solution

use The Law of Sines to solve for angle C
use the Sum of Angles Rule to find the other angle, B
use The Law of Sines to solve for the last side, b
Example:

If a < c we have 3 potential situations. "If sin(A) < a/c, there are two possible triangles satisfying the given conditions. If sin(A) = a/c, there is one possible triangle. If sin(A) > a/c, there are no possible triangles." ^[1]

sin(A) < a/c, there are two possible triangles

solve for the 2 possible values of the 3rd side b = c*cos(A) ± √[ a² - c² sin² (A) ]^[1]

for each set of solutions, use The Law of Cosines to solve for each of the other two angles

present 2 full solutions

Example:
triangle ass theorem with 2 solutions

sin(A) = a/c, there is one possible triangle

use The Law of Sines to solve for an angle, C

use the Sum of Angles Rule to find the other angle, B

use The Law of Sines to solve for the last side, b

Example:
triangle ass theorem with 1 solution

sin(A) > a/c, there are no possible triangles

Error Notice: sin(A) > a/c so there are no solutions and no triangle!

Example:
triangle ass theorem with 0 solutions

SAS is Side, Angle, Side

Given the size of 2 sides (c and a) and the size of the angle B that is in between those 2 sides you can calculate the sizes of the remaining 1 side and 2 angles.

use The Law of Cosines to solve for the remaining side, b

determine which side, a or c, is smallest and use the Law of Sines to solve for the size of the opposite angle, A or C respectively.^[2]

use the Sum of Angles Rule to find the last angle

SSS is Side, Side, Side

Given the sizes of the 3 sides you can calculate the sizes of all 3 angles in the triangle.

use The Law of Cosines to solve for the angles. You could also use the Sum of Angles Rule to find the final angle once you know 2 of them.

Sum of Angles in a Triangle

In Degrees A + B + C = 180°

In Radians A + B + C = π

Law of Sines

If a, b and c are the lengths of the legs of a triangle opposite to the angles A, B and C respectively; then the law of sines states:

a/sin A = b/sin B = c/sin C

Solving, for example, for an angle, A = sin^-1 [ a*sin(B) / b ]

Law of Cosines

If a, b and c are the lengths of the legs of a triangle opposite to the angles A, B and C respectively; then the law of cosines states:

a² = c² + b² - 2bc cos A, solving for cos A, cos A = ( b² + c² - a² ) / 2bc

b² = a² + c² - 2ca cos B, solving for cos B, cos B = ( c² + a² - b² ) / 2ca

c² = b² + a² - 2ab cos C, solving for cos C, cos C = ( a² + b² - c² ) / 2ab

Solving, for example, for an angle, A = cos^-1 [ ( b² + c² - a² ) / 2bc ]

Other Triangle Characteristics

Triangle perimeter, P = a + b + c

Triangle semi-perimeter, s = 0.5 * (a + b + c)

Triangle area, K = √[ s*(s-a)*(s-b)*(s-c)]

Radius of inscribed circle in the triangle, r = √[ (s-a)*(s-b)*(s-c) / s ]

Radius of circumscribed circle around triangle, R = (abc) / (4K)

References/ Further Reading

[1] Weisstein, Eric W. "ASS Theorem." From MathWorld-- A Wolfram Web Resource. ASS Theorem.

[2] Math is Fun - Solving SAS Triangles

Zwillinger, Daniel (Editor-in-Chief). CRC Standard Mathematical Tables and Formulae, 31st Edition New York, NY: CRC Press, p. 512, 2003.

Weisstein, Eric W. "Triangle Properties." From MathWorld-- A Wolfram Web Resource. Triangle Properties.

Math is Fun at Solving Triangles.