Triangle Calculator - Solve Sides, Angles, Area & Perimeter Instantly

Solve triangles and calculate all properties including sides, angles, area, perimeter, base, and dimensions. Includes formulas and step-by-step solutions.

Length Unit:

Unit

Angle Unit:

Angle

All inputs and outputs use the selected units

Quick Modes

Find Missing Side

Calculate Angles

Find Area

Right Triangle

Input Method

Side-Side-Side

Enter Measurements

Side 1 (cm)

Side 2 (cm)

Side 3 (cm)

Important Disclaimer

These calculators/tools are provided for informational and educational purposes only. Results are estimates and approximations that may contain errors, inaccuracies, or omissions. Do not rely on these calculations for any critical, financial, legal, engineering, medical, health, or other professional decisions.

All results are provided "as is" and "as available" without any warranties, express or implied, including but not limited to warranties of merchantability, fitness for a particular purpose, accuracy, completeness, or non-infringement
Calculations may contain errors, bugs, or inaccuracies and are subject to change without notice
Always verify all calculations independently using multiple sources and methods
Consult with qualified, licensed professionals (accountants, engineers, lawyers, doctors, physicians, registered dietitians, nutritionists, fitness professionals, etc.) for any decisions that may have legal, financial, safety, or health implications. Health and fitness calculators are not a substitute for professional medical advice, diagnosis, or treatment.
The accuracy of results depends entirely on input data, assumptions, and algorithms, which may be incorrect or outdated
We disclaim all liability for any damages, losses, costs, or expenses of any kind arising from or related to the use of these calculators, including but not limited to direct, indirect, incidental, consequential, special, or punitive damages
You assume full responsibility and all risk for any decisions, actions, or consequences resulting from the use of these calculators

By using these calculators/tools, you acknowledge that you have read, understood, and agree to this disclaimer. If you do not agree, do not use these calculators/tools. For any professional, commercial, or critical applications, you must consult with certified experts and refer to applicable laws, regulations, standards, and industry best practices.

Solve triangles and find all their properties from various input combinations. This comprehensive tool helps you determine missing sides, angles, area, perimeter, heights, medians, angle bisectors, and radii. Whether you're working on geometry homework, planning construction projects, or analyzing survey data, you'll get accurate results with clear explanations of the mathematical methods used, including step-by-step solutions and formulas.

How to Use Triangle Calculator

Choose Input Method

Select how you want to provide information about your triangle. You can use Side-Angle-Side (SAS), Side-Side-Side (SSS), Angle-Side-Angle (ASA), Angle-Angle-Side (AAS), or Side-Side-Angle (SSA). There's also a special mode for right triangles and an option to enter vertex coordinates. Each method requires different information, so pick the one that matches what you know.

Enter Your Measurements

Fill in the required values based on your chosen input method. Use consistent units for all measurements. Angles can be entered in degrees or radians, and you can switch between them. The tool validates your entries and will notify you if any values need correction or if the triangle configuration is impossible.

Review Results

After calculating, you'll see all triangle properties organized into sections. Basic measurements include all three sides and angles. Additional calculations show area, perimeter, heights, medians, angle bisectors, inradius, and circumradius. The tool also classifies your triangle by sides and angles, identifying special types like equilateral, isosceles, right, acute, or obtuse.

Calculator Features

🔺

Multiple Input Methods

Solve triangles from various input combinations including SAS, SSS, ASA, AAS, SSA, right triangle, and coordinates

📐

Complete Property Calculations

Get all triangle properties: sides, angles, area, perimeter, heights, medians, angle bisectors, and radii

📏

Triangle Classification

Automatically identifies triangle type by sides (equilateral, isosceles, scalene) and angles (acute, right, obtuse)

✅

Input Validation

Checks triangle inequality, angle sum, and other geometric constraints with helpful error messages

🎯

Ambiguous Case Handling

Properly handles the SSA ambiguous case, showing when zero, one, or two solutions exist

📋

Copy Results

One-click copy to clipboard for any calculated value with visual confirmation

📚

Step-by-Step Solutions

See the mathematical methods used, including Law of Sines, Law of Cosines, and other formulas

📱

Mobile-Friendly Design

Optimized interface that works perfectly on phones, tablets, and desktop computers

Complete Function List

Seven different input methods for solving triangles:
Side-Angle-Side (SAS) solving:
Side-Side-Side (SSS) solving:
Angle-Side-Angle (ASA) solving:
Angle-Angle-Side (AAS) solving:
Side-Side-Angle (SSA) solving with ambiguous case handling:
Right triangle solving with multiple input combinations:
Coordinate-based triangle solving:
All three sides calculated:
All three angles in degrees and radians:
Area calculated using multiple methods for validation:
Perimeter and semi-perimeter:
Three heights or altitudes:
Three medians:
Three angle bisectors:
Inradius calculation:
Circumradius calculation:
Triangle classification by sides:
Triangle classification by angles:
Special property identification:
Input validation with clear error messages:
Triangle inequality checking:
Angle sum validation:
Copy to clipboard functionality:
Responsive mobile-first design:
Step-by-step solution display:

Common Calculations & Examples

Example 1: Solve Triangle Using Side-Side-Side

Problem: Find all angles and properties of a triangle with sides measuring 5, 6, and 7 units

Steps:

Select "Side-Side-Side (SSS)" input method
Enter side 1: 5
Enter side 2: 6
Enter side 3: 7
Click "Calculate"

Result: Angles: 44.4°, 57.1°, 78.5°; Area: 14.70 square units; Perimeter: 18 units

Explanation: Using the Law of Cosines, we can find each angle. The tool calculates angle A opposite side a, angle B opposite side b, and angle C opposite side c. The area is found using Heron's formula, which uses the semi-perimeter. The perimeter is simply the sum of all three sides. This method is useful when you only have side measurements, such as when surveying land or working with structural measurements.

Example 2: Solve Right Triangle from Two Sides

Problem: A right triangle has legs measuring 3 and 4 units. Find the hypotenuse and all angles

Steps:

Select "Right Triangle" input method
Choose "Two Sides" option
Enter first leg: 3
Enter second leg: 4
Click "Calculate"

Result: Hypotenuse: 5 units; Angles: 36.9°, 53.1°, 90°; Area: 6 square units

Explanation: The hypotenuse is found using the Pythagorean theorem: √(3² + 4²) = √25 = 5. The angles are calculated using trigonometry. The area is simply (3 × 4) / 2 = 6 square units. This is the classic 3-4-5 right triangle pattern, one of the most common Pythagorean triples used in construction and surveying.

Example 3: Solve Triangle Using Angle-Side-Angle

Problem: A triangle has angles of 30° and 45°, with the included side measuring 10 units

Steps:

Select "Angle-Side-Angle (ASA)" input method
Enter first angle: 30
Enter side: 10
Enter second angle: 45
Click "Calculate"

Result: Third angle: 105°; Sides: 5.2, 7.3, 10 units; Area: 18.9 square units

Explanation: The third angle is found by subtracting the sum of the two known angles from 180°, giving 105°. The Law of Sines is then used to find the remaining sides. The area can be calculated using the formula (1/2) × a × b × sin(C), where C is the included angle. This method is particularly useful in navigation and surveying when you can measure angles directly but have limited side measurements.

Example 4: Solve Triangle from Coordinates

Problem: Find the properties of a triangle with vertices at (0,0), (4,0), and (2,3)

Steps:

Select "Coordinates" input method
Enter vertex 1: x=0, y=0
Enter vertex 2: x=4, y=0
Enter vertex 3: x=2, y=3
Click "Calculate"

Result: Sides: 4, 3.61, 3.61 units; Angles: 67.4°, 56.3°, 56.3°; Area: 6 square units

Explanation: The tool calculates side lengths using the distance formula between each pair of vertices. Once all sides are known, the Law of Cosines determines each angle. The area is found using the shoelace formula, which gives 6 square units. This method is valuable when working with coordinate geometry, computer graphics, or GPS coordinates.

Example 5: Calculate the Base of a Triangle

Problem: A triangle has sides of 5, 6, and 7 units. Find the base and calculate area using base-height method

Steps:

Select "Side-Side-Side (SSS)" input method
Enter side 1: 5
Enter side 2: 6
Enter side 3: 7
Click "Calculate"
View the "Base Calculation" section in results

Result: Base: c = 7 units (longest side); Height: hc = 4.2 units; Area: 14.7 square units

Explanation: The base is typically the longest side of the triangle. In this case, side c (7 units) is the base. The tool calculates the corresponding height (hc = 4.2 units) and shows that the area can be calculated as ½ × 7 × 4.2 = 14.7 square units. You can use any side as the base - just ensure you use the height that corresponds to that base (the perpendicular distance from the opposite vertex).

How to calculate the base of a triangle?

The base of a triangle is any side you choose to use for area calculations. Typically, the longest side is used as the base. To calculate the base, you need to know which side you're using. The tool automatically identifies the base and shows the corresponding height. The area formula using base and height is: Area = ½ × base × height. You can use any side as the base - just make sure to use the height that corresponds to that base (the perpendicular distance from the opposite vertex to that side).

What formulas are used in triangle calculations?

The triangle calculator uses several key formulas: Law of Sines (a/sin(A) = b/sin(B) = c/sin(C)) for finding sides or angles when you have angle-side combinations, Law of Cosines (c² = a² + b² - 2ab×cos(C)) for finding sides or angles from side-side-angle or side-side-side combinations, Heron's formula for area from three sides, Pythagorean theorem for right triangles, and the angle sum formula (A + B + C = 180°). The tool displays all formulas used in your specific calculation.

How to use trigonometry to solve triangles?

Trigonometry is essential for solving triangles. Use Law of Sines when you have two angles and a side (AAS/ASA) or two sides and a non-included angle (SSA). Use Law of Cosines when you have three sides (SSS) or two sides and the included angle (SAS). For right triangles, use the Pythagorean theorem (a² + b² = c²) and trigonometric ratios (sin, cos, tan). The tool shows all trigonometric function values (sine, cosine, tangent) for each angle and explains when to use each method.

What is the triangle inequality?

The triangle inequality states that the sum of any two sides must be greater than the third side. For example, if sides are 3, 4, and 8, this doesn't work because 3 + 4 = 7, which is less than 8. The tool checks this automatically and will warn you if your measurements don't satisfy this requirement.

Why does SSA sometimes give two solutions?

The Side-Side-Angle combination creates an ambiguous case because the given information doesn't always uniquely determine the triangle. When the known side opposite the given angle is shorter than the other known side, two different triangles can satisfy the conditions. The tool shows both solutions when this occurs.

How accurate are the calculations?

The tool uses standard trigonometric formulas with high precision. Results are typically accurate to at least six decimal places. Multiple calculation methods are used where possible to verify consistency, such as calculating area using both Heron's formula and the base-height method.

Can I use this for non-Euclidean geometry?

This tool uses Euclidean geometry formulas, including the Law of Sines and Law of Cosines, which assume a flat plane. For spherical or hyperbolic geometry, different formulas would be needed. Most everyday applications use Euclidean geometry.

What is the difference between height and median?

A height (altitude) is a line drawn from a vertex perpendicular to the opposite side. A median connects a vertex to the midpoint of the opposite side. Both are useful measurements in geometry. The tool calculates all three heights and all three medians for your triangle.

How do I find the area if I only know the sides?

Use Heron's formula, which is available through the Side-Side-Side (SSS) input method. This formula uses the semi-perimeter and all three side lengths. The tool automatically applies Heron's formula when you provide three sides, making it the standard method for this situation.

What are inradius and circumradius?

The inradius is the radius of the largest circle that fits inside the triangle, tangent to all three sides. The circumradius is the radius of the circle that passes through all three vertices. Both are useful in advanced geometry and engineering calculations. The tool calculates both automatically.

Understanding Triangle Solving Methods

Different combinations of sides and angles require different solving methods. The Law of Sines states that the ratio of a side to the sine of its opposite angle is constant: a/sin(A) = b/sin(B) = c/sin(C). The Law of Cosines relates sides and included angles: c² = a² + b² - 2ab×cos(C). These laws form the foundation for solving triangles. When you have three sides (SSS), the Law of Cosines finds all angles. With two angles and a side (AAS or ASA), the Law of Sines determines the remaining sides.

Triangle Properties and Measurements

Triangles have many interesting properties. The area can be found using base times height divided by two, Heron's formula for three sides, or the formula (1/2)×a×b×sin(C) for two sides and an included angle. Heights (altitudes) extend from vertices perpendicular to opposite sides. Medians connect vertices to the midpoint of opposite sides, and all three medians intersect at the centroid. Angle bisectors divide angles in half and intersect at the incenter. The inradius relates to area and semi-perimeter, while the circumradius connects to the sides and area through R = abc/(4A).

Triangle Classification

Triangles are classified by sides and angles. By sides: equilateral triangles have all three sides equal, isosceles triangles have two equal sides, and scalene triangles have no equal sides. By angles: acute triangles have all angles less than 90 degrees, right triangles have one 90-degree angle, and obtuse triangles have one angle greater than 90 degrees. These classifications help identify special properties and simplify calculations. For example, equilateral triangles have all angles equal to 60 degrees, and isosceles triangles have two equal angles opposite the equal sides.

The Ambiguous Case of SSA

When solving triangles using Side-Side-Angle (SSA), sometimes there are two valid solutions, sometimes one, and sometimes none. This happens because the given information doesn't always uniquely determine the triangle. When the side opposite the known angle is shorter than the other known side, and the known angle is acute, two different triangles can satisfy the conditions. The tool handles this ambiguity by showing both solutions when they exist, helping you understand all possibilities for your triangle configuration.

Triangle Calculator - Solve Sides, Angles, Area & Perimeter Instantly

Quick Modes

Input Method

Enter Measurements

How to Use Triangle Calculator

Choose Input Method

Enter Your Measurements

Review Results

Calculator Features

Multiple Input Methods

Complete Property Calculations

Triangle Classification

Input Validation

Ambiguous Case Handling

Copy Results

Step-by-Step Solutions

Mobile-Friendly Design

Complete Function List

Common Calculations & Examples

Example 1: Solve Triangle Using Side-Side-Side

Example 2: Solve Right Triangle from Two Sides

Example 3: Solve Triangle Using Angle-Side-Angle

Example 4: Solve Triangle from Coordinates

Example 5: Calculate the Base of a Triangle

Tips for Using Triangle Calculator

Frequently Asked Questions

Understanding Triangle Calculator

Related Calculators & Tools

Area & Perimeter Calculator

Scientific Calculator

Unit Circle Calculator