Triangle Calculator

Enter Known Values

Input the values you know about your triangle. You must provide at least three pieces of information (excluding right angle if applicable). The calculator accepts various combinations: three sides, two sides and one angle, or two angles and one side.

Select Calculation Type

Choose the appropriate calculation method based on your known values. For right triangles with two sides, use the Pythagorean theorem. For other triangles, use Law of Sines or Law of Cosines as appropriate.

View Complete Solution

The calculator displays all calculated values including unknown sides, angles, area, and perimeter. Each result shows the formula used and step-by-step calculation details.

Interior Angles

The sum of interior angles in any triangle equals 180 degrees:

This fundamental property allows you to find a missing angle when you know the other two.

Area Calculations

Pythagorean Theorem (Right Triangles)

For right triangles, the relationship between sides is:

Where c is the hypotenuse (the side opposite the right angle). This formula finds any side when you know the other two.

Law of Sines

The Law of Sines relates sides and their opposite angles:

This formula solves triangles when you know either two angles and one side (AAS/ASA) or two sides and an angle opposite one of them.

Law of Cosines

The Law of Cosines generalizes the Pythagorean theorem:

This formula solves triangles when you know two sides and their included angle (SAS) or all three sides (SSS).

Perimeter

The perimeter is simply the sum of all three sides:

By Sides

By Angles

Architecture and Construction

Triangles provide structural stability in buildings, bridges, and towers. The triangle's inherent rigidity makes it the preferred shape for structural support. Engineers calculate triangle dimensions to ensure buildings can withstand forces.

Surveying and Navigation

Surveyors use triangle properties to measure distances. By establishing one measured base line and taking angle measurements to a point, they can calculate distances using triangulation. This method has been used for centuries to map land and navigate oceans.

Computer Graphics

Three-dimensional graphics rely heavily on triangles. Complex shapes are decomposed into triangular meshes, and triangle properties determine how light and shadow appear on surfaces.

Sports Fields

Many sports use triangular configurations for positioning and strategy. In soccer, for example, triangular passing patterns create effective ball movement.

Everyday Life

From determining what size table fits in a room (using diagonal measurement) to calculating material needed for a triangular garden bed, triangle calculations appear in countless daily situations.

Choosing the Right Formula

Match your known values to the appropriate formula. If you know two sides and the angle between them, use the SAS area formula or Law of Cosines. If you know two angles and one side, use Law of Sines.

Checking Your Results

Use the angle sum property (180°) to verify your calculated angles are correct. Also verify that the largest side is opposite the largest angle.

Precision Considerations

Trigonometric calculations produce approximate results. For construction or engineering, maintain appropriate precision and account for measurement uncertainties.

Example 1: Finding Area of a Scalene Triangle

Given a triangle with sides a = 7, b = 9, and c = 12, find the area.

First, calculate the semiperimeter: s = (7 + 9 + 12) / 2 = 14

Apply Heron's formula: A = √(14 × (14-7) × (14-9) × (14-12)) = √(14 × 7 × 5 × 2) = √980 ≈ 31.3 square units

Example 2: Using Law of Sines

Given angle A = 45°, angle B = 60°, and side a = 10, find side b.

Using the Law of Sines: b / sin(60°) = 10 / sin(45°)

b = 10 × sin(60°) / sin(45°) = 10 × 0.866 / 0.707 ≈ 12.25

Example 3: Using Law of Cosines

Given sides a = 8, b = 6, and angle C = 120° between them, find side c.

c² = 8² + 6² - 2(8)(6)cos(120°) = 64 + 36 - 96(-0.5) = 100 + 48 = 148, so c ≈ 12.17

Example 4: Finding All Angles from Three Sides

Given sides a = 5, b = 7, c = 9, find all angles.

Find angle A: cos(A) = (b² + c² - a²) / (2bc) = (49 + 81 - 25) / (2 × 7 × 9) = 105/126 = 0.8333, so A ≈ 33.6°

Find angle B: cos(B) = (a² + c² - b²) / (2ac) = (25 + 81 - 49) / (2 × 5 × 9) = 57/90 = 0.6333, so B ≈ 50.7°

Angle C = 180° - 33.6° - 50.7° = 95.7°

Incorrect Angle Sum

Remember that all three interior angles must sum to exactly 180°. A common error is adding only two angles and assuming the third is the remainder without verifying the total.

Side Length Validity

Before calculating, always verify that the three sides can actually form a triangle. The Triangle Inequality Theorem states that the sum of any two sides must be greater than the third side. If this condition fails, no triangle exists.

Ambiguous Case Confusion

When applying the Law of Sines with two sides and a non-included angle, remember that this can produce the "ambiguous case" with zero, one, or two valid solutions. Always check the height (h = b × sin(A)) against the known side length to determine how many solutions exist.

Mixing Radians and Degrees

The calculator operates in degrees. If your inputs or required outputs need radians, convert using the formula: radians = degrees × π/180. Confusing these units will produce completely incorrect results.

Trigonometric Surveying

In land surveying, triangulation combines multiple triangles to measure large distances. Surveyors create a network of known points and measure angles between them, then calculate distances using triangle relationships. This method was essential in mapping continents and establishing national boundaries.

Navigation Systems

Global Positioning System (GPS) technology relies on triangulation. Satellites act as reference points, and the device calculates its position by measuring angles to multiple satellites simultaneously. The mathematics behind GPS involves solving triangles in three-dimensional space.

Structural Engineering

Engineers use triangle calculations to analyze forces on bridges, towers, and buildings. By decomposing complex structures into triangular elements, they can calculate stress, load capacity, and stability. This approach ensures structural safety and material efficiency.

Computer-Aided Design

CAD software uses triangle meshes to represent three-dimensional objects. Each triangle has defined vertices and normals that determine how surfaces appear and interact with light. The rendering pipeline performs millions of triangle calculations per second to produce realistic images.

Common Triangle Side Ratios

Area Multipliers for Common Angles

Invalid Combinations

Some combinations of values cannot form a valid triangle. For example, if the sum of two sides is less than or equal to the third side, no triangle exists.

Ambiguous Case

When using Law of Sines with two sides and an angle opposite one of them, there may be zero, one, or two possible triangles. The calculator will indicate when this occurs.

Angle Measurement

The calculator typically works in degrees. If you need radians, conversion is necessary: radians = degrees × π/180.