Triangle Angle Calculator

Calculate all three angles of a triangle from its sides using the Law of Cosines, or find the missing angle from two known angles.

What is the Triangle Angle Calculator?

The Triangle Angle Calculator finds all three interior angles of any triangle using one of two methods: the Law of Cosines when all three side lengths are known (SSS), or the angle sum property when two of the three angles are already known. Both methods are fundamental results of Euclidean geometry and give exact answers for any valid triangle.

When you enter three side lengths, the calculator applies the Law of Cosines formula — cos A = (b² + c² − a²) / (2bc) — for each vertex in turn. This formula generalises the Pythagorean theorem: when the angle is 90°, the cosine term vanishes and the equation reduces to the familiar a² + b² = c². Once two angles are computed via the Law of Cosines, the third is obtained from the angle sum property: A + B + C = 180°. This keeps rounding error to a minimum by computing only two arccosines.

The calculator also classifies the triangle in two dimensions. By angles: if all three are less than 90° it is acute; if one equals 90° it is a right triangle; if one exceeds 90° it is obtuse. By sides: if all three sides are equal it is equilateral (all angles 60°); if exactly two sides are equal it is isosceles (with two equal base angles); if no sides are equal it is scalene. The combined label — such as “Scalene Right” or “Isosceles Acute” — tells you the full geometric character of the triangle at a glance.

Practical applications appear across many fields. Surveyors use the Law of Cosines to compute angles between measured distances in land triangulation. Architects and carpenters calculate roof pitch angles and mitre cut angles from known rafter lengths. Navigation and aviation use triangle angle calculations in bearing problems. Game developers apply them when rotating objects in 2D space. Students use this tool to verify manual calculations in trigonometry coursework, understand how the formula behaves for different triangle shapes, and build intuition about the relationship between sides and angles.

Formula — Law of Cosines and Angle Sum

Given a triangle with sides a, b, c opposite angles A, B, C respectively, the Law of Cosines gives each angle directly from the three side lengths:

cos A = (b² + c² − a²) / (2bc) → A = arccos((b² + c² − a²) / (2bc))

A = Angle opposite side a (in degrees)

b, c = The two sides adjacent to angle A

a = The side opposite to angle A

cos B = (a² + c² − b²) / (2ac) → B = arccos((a² + c² − b²) / (2ac))

B = Angle opposite side b (in degrees)

a, c = The two sides adjacent to angle B

C = 180° − A − B (Angle Sum Property)

C = Third angle, derived once A and B are known

The interior angles of any Euclidean triangle always sum to exactly 180°

For the Two Angles mode, only the angle sum property is needed: if two angles are known, the third is simply their difference from 180°. The triangle inequality must hold for any valid triangle: a + b > c, a + c > b, and b + c > a. This calculator validates the inequality before computing and alerts you if it is violated.

How to Use This Calculator

Choose your input mode — Select Three Sides (SSS) if you know all three side lengths, or Two Angles if you know two of the three angles and want the third.
Enter the values — In SSS mode, type the three side lengths in any consistent unit (cm, m, inches). They must satisfy the triangle inequality. In Two Angles mode, enter any two angles in degrees — both must be positive and their sum must be less than 180.
Click Calculate — The calculator applies the Law of Cosines (SSS mode) or angle sum property (Two Angles mode) and instantly shows all three angles, triangle classification, and perimeter.
Try a worked example — Scroll to the Examples section and click any Try this example link to auto-fill real values and see the step-by-step working in the note box below the results.

Example Calculations

Example 1 — Classic 3-4-5 Right Triangle

Sides a = 3, b = 4, c = 5 (SSS mode)

cos A = (b² + c² − a²) / (2bc) = (16 + 25 − 9) / (2 × 4 × 5) = 32 / 40 = 0.8

A = arccos(0.8) = 36.8699°

cos B = (9 + 25 − 16) / (2 × 3 × 5) = 18 / 30 = 0.6 → B = arccos(0.6) = 53.1301°

C = 180 − 36.8699 − 53.1301 = 90.0000° — confirms this is a right triangle

A = 36.8699° · B = 53.1301° · C = 90.0000° · Scalene Right · Perimeter = 12

Try this example →

Example 2 — Equilateral Triangle (5-5-5)

Sides a = 5, b = 5, c = 5 (SSS mode)

cos A = (25 + 25 − 25) / (2 × 5 × 5) = 25 / 50 = 0.5

A = arccos(0.5) = 60.0000°

By symmetry (all sides equal), B = 60.0000° and C = 180 − 60 − 60 = 60.0000°

A = 60.0000° · B = 60.0000° · C = 60.0000° · Equilateral Acute · Perimeter = 15

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Example 3 — Isosceles Triangle (6-6-10)

Sides a = 6, b = 6, c = 10 (SSS mode)

cos A = (36 + 100 − 36) / (2 × 6 × 10) = 100 / 120 ≈ 0.8333 → A = arccos(0.8333) ≈ 33.5573°

cos B = (36 + 100 − 36) / (2 × 6 × 10) = 0.8333 → B ≈ 33.5573° (equal to A since a = b)

C = 180 − 33.5573 − 33.5573 = 112.8854° — one angle > 90°, so obtuse

A ≈ 33.5573° · B ≈ 33.5573° · C ≈ 112.8854° · Isosceles Obtuse · Perimeter = 22

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Example 4 — Two Angles Known (45° + 75°)

Angle 1 = 45°, Angle 2 = 75° (Two Angles mode)

Third angle = 180 − 45 − 75 = 60° (angle sum property)

All three angles are less than 90°, so the triangle is Acute

The sides are not known, so the triangle is classified by angle type only. Side lengths would require the Law of Sines with a known reference side.

Angle A = 45.0000° · Angle B = 75.0000° · Angle C = 60.0000° · Acute

Try this example →

Frequently Asked Questions

How do you find the angles of a triangle from its sides?+

Use the Law of Cosines: cos A = (b² + c² − a²) / (2bc). Solve for angle A in degrees: A = arccos((b² + c² − a²) / (2bc)). Repeat for B and C. The three angles must sum to 180°. Example: sides 3, 4, 5 → A = arccos((16+25−9)/40) = arccos(32/40) = arccos(0.8) = 36.87°; B = arccos((9+25−16)/30) = 53.13°; C = 90°.

What is the Law of Cosines?+

The Law of Cosines generalises the Pythagorean theorem to any triangle: c² = a² + b² − 2ab·cos(C). It can be rearranged to find any angle: cos(C) = (a² + b² − c²) / (2ab). When C = 90°, cos(C) = 0, and the formula reduces to the Pythagorean theorem: c² = a² + b². The Law of Cosines works for all triangles - acute, right, and obtuse.

How do you find the missing angle of a triangle when two angles are known?+

Use the angle sum property: A + B + C = 180°. If you know angles A and B, then C = 180° − A − B. Example: if two angles are 45° and 75°, the third is 180° − 45° − 75° = 60°. This is the simplest case and does not require any knowledge of the side lengths.

What are acute, right, and obtuse triangles?+

Acute triangle: all three angles less than 90°. Right triangle: exactly one angle equals 90° (and a² + b² = c² for sides). Obtuse triangle: exactly one angle greater than 90°. To classify from side lengths: if the largest side c satisfies c² < a² + b², it is acute; c² = a² + b², it is right; c² > a² + b², it is obtuse.

What is the angle sum property of a triangle?+

The interior angles of any triangle always sum to exactly 180°. This is a fundamental theorem of Euclidean geometry. It follows from the fact that a straight line has angle 180°, and the three angles of a triangle can always be rearranged to form a straight line. In non-Euclidean geometry (on curved surfaces), this sum can differ from 180°.

How does the Law of Cosines relate to the Pythagorean theorem?+

The Pythagorean theorem is a special case of the Law of Cosines. The Law of Cosines states c² = a² + b² − 2ab·cos(C). When C = 90°, cos(90°) = 0, so the −2ab·cos(C) term vanishes, leaving c² = a² + b² - exactly the Pythagorean theorem. The Law of Cosines extends this to any angle C.

What is an isosceles triangle and what are its angle properties?+

An isosceles triangle has two equal sides. The angles opposite the two equal sides (the base angles) are also equal. For example, if sides a = b, then angles A = B. If all three sides are equal (equilateral), then A = B = C = 60°. This property allows isosceles triangles to be solved with less information than a general scalene triangle.

Can I find all three angles if I only know two sides?+

Not uniquely. Two sides alone do not determine a triangle - you also need either the included angle (SAS) or the third side (SSS). If you know two sides and the included angle, use the Law of Cosines to find the third side, then find the remaining angles. If you know two sides and the angle opposite one of them (SSA), two triangles may be possible (the ambiguous case).

What is the Law of Sines and when should I use it instead?+

Law of Sines: a/sin(A) = b/sin(B) = c/sin(C). Use it when you know: (1) two angles and any side (AAS or ASA), or (2) two sides and an angle opposite one of them (SSA - but beware the ambiguous case). Use the Law of Cosines when you know: (1) all three sides (SSS), or (2) two sides and the included angle (SAS). Both laws complement each other for solving triangles.

What is the exterior angle of a triangle?+

An exterior angle of a triangle is formed by extending one side beyond the vertex. The exterior angle equals the sum of the two non-adjacent interior angles (remote interior angles). If the interior angles at vertices A and B are α and β, the exterior angle at C = α + β. The exterior angle is always greater than either remote interior angle. The sum of all three exterior angles of a triangle = 360°.

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📌 Quick Tips

💡The angles of any triangle always sum to exactly 180°. Use this to verify your results.

💡A right triangle has exactly one 90° angle. If the Law of Cosines gives cos A = 0, angle A = 90°.

💡An obtuse triangle has one angle greater than 90°. The longest side is always opposite the largest angle.

💡For an equilateral triangle (all sides equal), all angles are exactly 60°.

💡The triangle inequality must hold: each side must be less than the sum of the other two. This calculator validates this automatically.