Maths⏱ 5 min read
How to Calculate Angles in Any Triangle
The angles of a triangle always sum to 180°. But finding unknown angles in non-right triangles requires the sine and cosine rules. Here's a complete guide with worked examples.
Triangle angle calculations range from trivially simple (all three angles given, check they sum to 180°) to genuinely tricky (two sides and a non-included angle). Here's every case you'll encounter.
The Fundamental Rule
Angles in any triangle always sum to 180°.
If two angles are known, the third is:
Angle C = 180° − Angle A − Angle B
Example: A = 45°, B = 72°
C = 180° − 45° − 72° = 63°
Right-Angled Triangles: SOHCAHTOA
sin(θ) = Opposite ÷ Hypotenuse
cos(θ) = Adjacent ÷ Hypotenuse
tan(θ) = Opposite ÷ Adjacent
To find angle θ: use inverse trig functions
θ = arcsin(O/H) = arccos(A/H) = arctan(O/A)
Example: right triangle, opposite = 5, adjacent = 8
tan(θ) = 5 ÷ 8 = 0.625
θ = arctan(0.625) = 32.0°
Check: third angle = 180 − 90 − 32 = 58°
The Sine Rule (Any Triangle)
Use when you know: two angles and one side (AAS/ASA), or two sides and a non-included angle (SSA — beware ambiguous case).
a/sin(A) = b/sin(B) = c/sin(C)
Where a, b, c are sides opposite angles A, B, C.
Example: A = 40°, B = 75°, a = 8cm. Find b.
C = 180 − 40 − 75 = 65°
b/sin(75°) = 8/sin(40°)
b = 8 × sin(75°)/sin(40°)
b = 8 × 0.9659/0.6428 = 12.02 cm
The Cosine Rule (Any Triangle)
Use when you know: three sides (SSS), or two sides and the included angle (SAS).
Finding a side:
a² = b² + c² − 2bc·cos(A)
Finding an angle (rearranged):
cos(A) = (b² + c² − a²) / (2bc)
A = arccos((b² + c² − a²) / (2bc))
Example: b = 7, c = 9, A = 50°. Find a.
a² = 7² + 9² − 2(7)(9)cos(50°)
= 49 + 81 − 126 × 0.6428
= 130 − 80.99 = 49.01
a = √49.01 = 7.00 cm
Example: a = 5, b = 7, c = 8. Find angle A.
cos(A) = (49 + 64 − 25)/(2×7×8) = 88/112 = 0.7857
A = arccos(0.7857) = 38.2°
Equilateral and Isosceles Shortcuts
Equilateral triangle: all angles = 60° always.
Isosceles triangle (two equal sides → two equal base angles):
If apex angle = X, base angles = (180 − X) ÷ 2
Example: apex angle 40° (isosceles)
Base angles = (180 − 40) ÷ 2 = 70° each
Check: 40 + 70 + 70 = 180° ✓
Exterior Angles
An exterior angle of a triangle equals the sum of the two non-adjacent interior angles.
If interior angles are A, B, C:
Exterior angle at C = A + B = 180° − C
Example: A = 55°, B = 70°, C = 55°
Exterior angle at B = 55 + 55 = 110°
Check: 180 − 70 = 110° ✓