Triangle Angle Calculator – Find Missing Angles in a Triangle
Find any missing angle in a triangle with our free online angle calculator. Enter known angles or sides and instantly solve for the remaining angles using trigonometry rules.
Enter two angles to find the third, or all three to verify.
Understanding Triangle Angles
Every triangle has three angles that always add up to 180 degrees. This fundamental rule – the Triangle Angle Sum Theorem – lets you find any missing angle when you know the other two. But triangles can be solved in multiple ways depending on what information you have.
This calculator handles three common scenarios: finding a missing angle from two known angles, calculating all angles when you know all three sides (using the Law of Cosines), or solving a triangle when you have two sides and the angle between them. Each method is shown step by step.
Triangle Angle Methods
AA (Angle-Angle)
Given two angles, subtract their sum from 180° to find the third. This is the simplest case and works for any triangle.
SSS (Side-Side-Side)
Given three sides, use the Law of Cosines to find each angle. The sides must satisfy the triangle inequality.
SAS (Side-Angle-Side)
Given two sides and the included angle, first find the third side using Law of Cosines, then use Law of Sines for remaining angles.
Types of Triangles by Angles
Acute Triangle
All three angles are less than 90°.
Right Triangle
One angle is exactly 90°. The other two angles are complementary (sum to 90°).
Obtuse Triangle
One angle is greater than 90°. Only one angle can be obtuse.
Equiangular Triangle
All three angles are equal. Each angle is exactly 60°.
Isosceles Triangle
At least two angles are equal. The equal sides have equal opposite angles.
Scalene Triangle
All three angles are different. All three sides have different lengths.
Key Formulas
Triangle Angle Sum Theorem
The foundation of triangle geometry. Use this when you know two angles and need the third.
Law of Cosines
Generalizes the Pythagorean theorem for any triangle. Use when you know all three sides (SSS) or two sides and the included angle (SAS).
Law of Sines
Relates angles to their opposite sides. Useful when you know one angle-side pair and want to find another angle or side.
Worked Examples
Example 1: Finding a missing angle
Given: Angle A = 65°, Angle B = 55°. Find Angle C.
C = 180° - A - B
C = 180° - 65° - 55°
C = 180° - 120°
C = 60°
The third angle is 60°. All three angles (65°, 55°, 60°) add up to 180°.
Example 2: Angles from three sides (SSS)
Given: a = 5, b = 7, c = 8. Find all angles.
Using Law of Cosines for Angle A:
cos(A) = (b² + c² - a²) / 2bc
cos(A) = (49 + 64 - 25) / (2 × 7 × 8)
cos(A) = 88 / 112 = 0.7857
A = arccos(0.7857) ≈ 38.21°
Similarly: B ≈ 60.00°, C ≈ 81.79°
Check: 38.21° + 60.00° + 81.79° = 180° ✓
Example 3: SAS triangle solution
Given: a = 6, b = 8, Angle C = 60°. Find the rest.
Step 1: Find side c
c² = 6² + 8² - 2(6)(8)·cos(60°)
c² = 36 + 64 - 96·0.5 = 100 - 48 = 52
c = √52 ≈ 7.21
Step 2: Find Angle A using Law of Sines
sin(A) = a·sin(C)/c = 6·sin(60°)/7.21 ≈ 0.7207
A = arcsin(0.7207) ≈ 46.10°
Step 3: Find Angle B
B = 180° - 60° - 46.10° = 73.90°
Complete triangle: A ≈ 46.10°, B ≈ 73.90°, C = 60°, c ≈ 7.21
Quick Fact
The Triangle Angle Sum Theorem (angles add to 180°) is only true in Euclidean geometry – the flat geometry we learn in school. On a sphere, like Earth's surface, triangle angles add up to MORE than 180°. A triangle drawn from the North Pole to two points on the equator has three right angles, totaling 270°! This "spherical geometry" is what pilots and sailors use for navigation.
Frequently Asked Questions
Why do triangle angles add to 180 degrees?
Draw a line parallel to one side through the opposite vertex. The alternate interior angles formed are equal to the triangle's base angles. These three angles form a straight line, which is 180°. This proof works for any triangle in Euclidean geometry.
Can a triangle have two right angles?
No. Two right angles would sum to 180°, leaving 0° for the third angle – impossible. A triangle can have at most one right angle or one obtuse angle.
What if I only know one angle?
One angle alone isn't enough to determine a unique triangle. You need at least one side length, or two angles, or two sides. With only one angle, infinitely many triangles are possible.
How do I find angles in a right triangle?
One angle is 90°. If you know one acute angle, subtract it from 90° to find the other. If you know sides, use trigonometric ratios: sin, cos, or tan, then use inverse functions.
What is the triangle inequality?
For three lengths to form a triangle, the sum of any two sides must exceed the third side. If a + b ≤ c (or any similar combination), those lengths cannot form a triangle.
When do I use Law of Sines vs Law of Cosines?
Use Law of Cosines for SSS (three sides) or SAS (two sides + included angle). Use Law of Sines for AAS, ASA, or SSA (when you have an angle-side pair and want to find another angle or side).
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