Triangle Angle Calculator | Find Missing Angles (SSS & AA)

What is a Triangle? Core Geometric Principles

A triangle is a fundamental polygon with three edges and three vertices.
The sum of the interior angles of any triangle always equals 180 degrees.
The relationship between sides and angles is governed by the Laws of Sines and Cosines.

A triangle is one of the most basic shapes in geometry, defined by three points (vertices) connected by three straight lines (sides or edges). Understanding triangles is crucial as they form the building blocks for more complex polygons and structures. The most fundamental property of a simple (non-self-intersecting) triangle is that the sum of its three internal angles is always 180°.

The Angle Sum Property

The rule that angles add up to 180° (A + B + C = 180°) is the cornerstone of solving for unknown angles. If you know any two angles in a triangle, you can instantly find the third. This property holds true for all types of triangles, whether they are equilateral, isosceles, or scalene.

The Law of Cosines

When you know the lengths of all three sides (an SSS triangle), you can't use the angle sum property directly. Instead, you must use the Law of Cosines. It relates the lengths of the sides to the cosine of one of its angles. The formula to find angle C is: C = arccos((a² + b² - c²) / 2ab). Similar formulas can be derived for angles A and B.

Fundamental Calculations

If Angle A = 50° and Angle B = 70°, then Angle C = 180° - 50° - 70° = 60°.
For a triangle with sides a=3, b=4, c=5, Angle C (opposite side c) is arccos((3²+4²-5²)/(2*3*4)) = arccos(0) = 90°.

Step-by-Step Guide to Using the Triangle Angle Calculator

Select the appropriate calculation method based on your known values.
Enter your data accurately into the designated fields.
Interpret the results, which include angles, triangle type, and area.

Our calculator simplifies the process of finding missing angles. Follow these steps for accurate results:

Step 1: Choose Your Calculation Method

Begin by selecting from the dropdown menu. Choose 'From Two Angles (AA)' if you know two angles. Choose 'From Three Sides (SSS)' if you know the lengths of all three sides.

Step 2: Input Your Values

For the AA method, enter the two known angles in degrees into the 'Angle A' and 'Angle B' fields. The values must be positive and their sum must be less than 180°.

For the SSS method, enter the lengths of the three sides into the 'Side a', 'Side b', and 'Side c' fields. The values must be positive and satisfy the Triangle Inequality Theorem (the sum of the lengths of any two sides of a triangle must be greater than the length of the third side).

Step 3: Calculate and Analyze

Click the 'Calculate Angles' button. The calculator will display the values for all three angles (A, B, and C). It will also classify the triangle (e.g., Acute, Obtuse, Right, Equilateral, Isosceles) and calculate its area using Heron's formula for the SSS case.

Practical Usage Examples

Input: Method=AA, Angle A=45, Angle B=90 → Result: Angle C=45, Type=Right Isosceles.
Input: Method=SSS, Side a=5, Side b=5, Side c=8 → Result: Angles ≈ 38.6°, 38.6°, 102.8°, Type=Obtuse Isosceles.

Real-World Applications of Triangle Angle Calculation

Architecture and Construction: Ensuring structural integrity and perfect angles.
Navigation and Surveying: Pinpointing locations using triangulation.
Physics and Engineering: Analyzing forces and vectors.

Calculating triangle angles is not just an academic exercise; it's a critical skill in numerous professional fields.

Architecture and Engineering

Architects use angle calculations to design stable and aesthetically pleasing structures, such as sloped roofs, trusses, and support beams. Civil engineers rely on these principles to ensure bridges and other infrastructure can safely distribute loads.

Land Surveying and Cartography

Surveyors use a technique called triangulation to determine distances and elevations. By measuring the angles from two known points to a third, unknown point, they can create accurate maps and property boundaries.

Computer Graphics and Game Development

In the digital world, triangles are the fundamental components of 3D models. Developers use angle calculations for lighting effects, object rotation, and collision detection to create realistic and immersive virtual environments.

Industry Applications

An architect designing an A-frame house needs to calculate the roof pitch angles for stability and water runoff.
A GPS satellite uses triangulation with signals from multiple ground stations to determine its precise location.
A game developer calculates the angle of reflection for a light ray bouncing off a surface.

Triangle Classification and Properties

Classification by Angles: Acute, Obtuse, and Right.
Classification by Sides: Equilateral, Isosceles, and Scalene.
Understanding the relationship between side lengths and angle measures.

Triangles can be classified based on their angles or their sides. Our calculator helps identify the type of triangle you are working with.

Classification by Angles

Acute Triangle: All three angles are less than 90°. For example, a triangle with angles 60°, 70°, 50°.

Obtuse Triangle: One angle is greater than 90°. For example, a triangle with angles 30°, 50°, 100°.

Right Triangle: One angle is exactly 90°. The sides of a right triangle are related by the Pythagorean theorem (a² + b² = c²).

Classification by Sides

Equilateral Triangle: All three sides are equal in length, and all three angles are equal (60° each).

Isosceles Triangle: Two sides are equal in length. The angles opposite the equal sides are also equal.

Scalene Triangle: All three sides have different lengths, and all three angles are different.

Triangle Type Examples

Sides 5, 12, 13 form a Right Scalene triangle.
Sides 7, 7, 7 form an Equilateral (and Acute) triangle.
Angles 20°, 40°, 120° form an Obtuse Scalene triangle.

Mathematical Formulas and Derivations

The Angle Sum Property: A + B + C = 180°.
The Law of Cosines: c² = a² + b² - 2ab cos(C).
Heron's Formula for Area from Sides.

The calculations performed by this tool are based on established geometric theorems.

Derivation for the 'From Two Angles' Method

This is the most straightforward derivation. Given that the sum of angles in a Euclidean triangle is always 180 degrees, if we know Angle A and Angle B, we can find Angle C by simple subtraction: C = 180° - A - B.

Derivation for the 'From Three Sides' Method

This method relies on the Law of Cosines. To find each angle, the formula is rearranged to solve for the angle itself. For example, to find Angle C:

1. Start with the Law of Cosines: c² = a² + b² - 2ab cos(C)

2. Isolate the cosine term: 2ab cos(C) = a² + b² - c²

3. Solve for cos(C): cos(C) = (a² + b² - c²) / 2ab

4. Take the inverse cosine (arccos) to find the angle: C = arccos((a² + b² - c²) / 2ab)

This process is repeated for the other two angles.

Area Calculation (Heron's Formula)

When all three sides are known, the area can be calculated using Heron's Formula. First, calculate the semi-perimeter (s): s = (a + b + c) / 2. Then, the area is: Area = √[s(s-a)(s-b)(s-c)].

Formula Application

Law of Cosines: For sides a=8, b=10, c=12, Angle C = arccos((8²+10²-12²)/(2*8*10)) = arccos(20/160) = arccos(0.125) ≈ 82.8°.
Heron's Formula: For a=8, b=10, c=12, s = (8+10+12)/2 = 15. Area = √[15(15-8)(15-10)(15-12)] = √[15*7*5*3] = √1575 ≈ 39.7.

Find Third Angle

Isosceles Triangle

Equilateral Triangle

Scalene Triangle (3-4-5)