Triangle Angle Calculator

The sum of all three interior angles in any triangle is always exactly 180° (or π radians). This is a fundamental theorem of Euclidean geometry and applies to all triangles — scalene, isosceles, and equilateral.

Subtract the two known angles from 180°: third angle = 180° - angle₁ - angle₂. For example, if two angles are 65° and 48°, the third is 180° - 65° - 48° = 67°.

Use the law of cosines: cos(A) = (b² + c² - a²) / (2bc). Compute A = arccos(...). Repeat for angle B, then find C = 180° - A - B. This works for any triangle where you know all three sides.

The law of cosines relates the sides and angles of any triangle: c² = a² + b² - 2ab cos(C). Rearranged: cos(C) = (a² + b² - c²) / (2ab). It generalizes the Pythagorean theorem (which applies when C = 90°).

The law of sines states: a/sin(A) = b/sin(B) = c/sin(C) = 2R, where R is the circumradius. It is useful when you know a side and its opposite angle, plus one other piece of information.

No. Since the angle sum is 180°, having two 90° angles would leave 0° for the third angle, which is impossible. A triangle can have at most one right angle (90°) or one obtuse angle (>90°).

An obtuse triangle has one angle greater than 90°. The other two angles must be acute (less than 90°) and their sum must be less than 90°. For example, a triangle with angles 100°, 45°, and 35° is obtuse.

An acute triangle has all three angles less than 90°. For example, an equilateral triangle (all angles 60°) is acute. The triangle with angles 70°, 60°, 50° is also acute.

In a right triangle (one 90° angle), the two acute angles are complementary: they add to 90°. Use trigonometry: angle = arcsin(opposite/hypotenuse) = arccos(adjacent/hypotenuse) = arctan(opposite/adjacent).

When you know two sides and a non-included angle (SSA), there may be 0, 1, or 2 possible triangles. This is because the given angle may allow the opposite side to swing to two different positions. The calculator handles this case.

Multiply degrees by π/180 to get radians. Multiply radians by 180/π to get degrees. For example: 60° = π/3 ≈ 1.047 radians, and π/4 radians = 45°.

An exterior angle equals the sum of the two non-adjacent interior angles. If interior angles are A, B, C, the exterior angle at C is A + B = 180° - C. This is a direct consequence of the angle sum property.

Yes. Using the area formula A = (1/2)ab sin(C), you can find sin(C) = 2A/(ab), then C = arcsin(2A/(ab)). Use the law of cosines for the other angles.

In any triangle, the largest angle is opposite the longest side, and the smallest angle is opposite the shortest side. Equal sides are opposite equal angles. This ordering is preserved across the triangle.

The angle bisector divides an angle into two equal parts. Its length from vertex A to the opposite side is: t_a = (2bc cos(A/2)) / (b + c). The bisector divides the opposite side in ratio b:c.

The orthocenter is the intersection of all three altitudes. In an acute triangle, it is inside the triangle. In an obtuse triangle, it is outside. In a right triangle, it is at the vertex of the right angle.

The incenter is the intersection of all three angle bisectors. It is the center of the inscribed circle (incircle), which touches all three sides. The incenter is always inside the triangle.

Surveyors use angle measurements and the law of sines/cosines to calculate distances between inaccessible points. Triangulation — measuring angles from known baselines — is the foundation of land surveying and GPS positioning.

Interior angles are inside the triangle (sum = 180°). Exterior angles are formed by extending one side (each exterior angle = 180° - adjacent interior angle). The sum of exterior angles (one at each vertex) is always 360°.

No. All interior angles of a triangle are positive and between 0° and 180° (exclusive). Negative angles or angles ≥ 180° are not valid for triangle interior angles.

The Triangle Angle Calculator uses standard mathematical formulas validated against authoritative references. The specific formula is displayed in the calculator interface with a detailed explanation of each variable.

Each calculator includes a step-by-step worked example showing exactly how the formula is applied. You can follow these steps with pen and paper to verify any result.

The Triangle Angle Calculator accepts numeric inputs including integers and decimals. Invalid inputs (letters, special characters) are rejected with clear error messages.