Perimeter of a Triangle Calculator

Calculate triangle perimeter with multiple methods

Select a calculation method and enter the required values to find the perimeter. The calculator also validates the inputs to ensure they form a valid triangle.

Examples

Click on an example to load it into the calculator.

Standard Triangle (3 Sides)

3 Side Lengths

A simple scalene triangle with sides 5, 6, and 7.

Side a: 5 m

Side b: 6 m

Side c: 7 m

SAS (Side-Angle-Side)

SAS (Side-Angle-Side)

Two sides (10, 12) and the angle (50°) between them.

Side a: 10 m

Side b: 12 m

Angle C (°): 50°

ASA (Angle-Side-Angle)

ASA (Angle-Side-Angle)

Two angles (45°, 60°) and the side (8) between them.

Side a: 8 m

Angle B (°): 45°

Angle C (°): 60°

Coordinates of Vertices

Coordinates of Vertices

A triangle defined by three points in a 2D plane.

Aₓ: 1

Aᵧ: 2

Bₓ: 7

Bᵧ: 2

Cₓ: 4

Cᵧ: 6

Other Titles
Understanding the Perimeter of a Triangle: A Comprehensive Guide
Master the calculation of a triangle's perimeter with various methods and explore its real-world applications, from construction to cartography.

What is the Perimeter of a Triangle?

  • Defining the perimeter as the total distance around a triangle.
  • The fundamental formula P = a + b + c.
  • The importance of the Triangle Inequality Theorem.
The perimeter of a triangle is the total length of its boundary. It is one of the most basic and essential properties of a polygon. To find the perimeter, you simply sum the lengths of its three sides. This concept applies to all types of triangles, including equilateral (all sides equal), isosceles (two sides equal), and scalene (no sides equal).
The Basic Formula
For a triangle with side lengths a, b, and c, the perimeter (P) is given by the straightforward formula: P = a + b + c.
Triangle Inequality Theorem
A crucial rule for any valid triangle is the Triangle Inequality Theorem. It states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side (a + b > c, a + c > b, and b + c > a). Our calculator automatically validates this to ensure the given dimensions are possible.

Fundamental Concepts

  • For a triangle with sides 3, 4, 5: P = 3 + 4 + 5 = 12. This is a valid triangle.
  • For sides 2, 3, 6: This is not a valid triangle because 2 + 3 = 5, which is not greater than 6.

Step-by-Step Guide to Using the Calculator

  • Choosing the right calculation method for your data.
  • Entering values for sides, angles, or coordinates.
  • Interpreting the detailed results provided.
This calculator is designed to be flexible, offering multiple methods to find a triangle's perimeter based on the information you have.
1. Using 3 Side Lengths
This is the most direct method. Simply input the lengths of the three sides (a, b, c) into their respective fields. The calculator will sum them up and check the Triangle Inequality Theorem.
2. Using 2 Sides and an Included Angle (SAS)
If you know two sides and the angle between them (e.g., sides a, b, and angle C), the calculator uses the Law of Cosines (c² = a² + b² - 2ab cos(C)) to find the third side first, then calculates the perimeter.
3. Using 2 Angles and an Included Side (ASA)
Given two angles and the side between them (e.g., angle B, side a, angle C), the calculator first finds the third angle (A = 180° - B - C). Then, it uses the Law of Sines (b/sin(B) = a/sin(A)) to determine the lengths of the other two sides before summing them for the perimeter.
4. Using 3 Vertex Coordinates
If you have the coordinates of the three vertices (A, B, C) in a Cartesian plane, the calculator applies the distance formula (d = √((x₂-x₁)² + (y₂-y₁)²)) between each pair of points to find the length of each side. It then adds these lengths to get the perimeter.

Method Walkthrough

  • SAS Example: sides=8, 10; angle=45°. The calculator finds the third side (~7.05) and then the perimeter (~25.05).
  • Coordinates Example: A(0,0), B(4,0), C(2,3). The calculator finds side lengths (~4, ~3.6, ~2.2) and perimeter (~9.8).

Real-World Applications

  • Applications in architecture, engineering, and construction.
  • Use in land surveying and cartography.
  • Importance in physics, navigation, and graphic design.
Calculating the perimeter of a triangle is not just a classroom exercise; it's a fundamental task in many professional and practical fields.
Construction and Engineering
Architects and engineers constantly calculate perimeters to determine the amount of material needed for fencing, framing, or creating structural supports. For instance, calculating the amount of fencing required for a triangular garden plot.
Land Surveying
Surveyors use triangle perimeters when mapping parcels of land. By dividing a complex plot into a series of triangles (a technique called triangulation), they can accurately measure its boundaries and area.
Navigation and Art
In navigation, sailors and pilots may use triangulation to determine their position. In art and design, understanding the perimeter of shapes is crucial for composition, framing, and material estimation.

Practical Scenarios

  • Fencing a triangular yard with sides 50m, 60m, and 70m requires 180m of fencing.
  • A truss in a bridge design forms a triangle; its perimeter is needed to calculate the total length of steel required.

Common Misconceptions and Correct Methods

  • Confusing perimeter with area.
  • Ignoring the Triangle Inequality Theorem.
  • Errors in applying the Laws of Sines and Cosines.
Perimeter vs. Area
A frequent mistake is confusing perimeter with area. The perimeter is the one-dimensional distance around the triangle, measured in units of length (e.g., meters). The area is the two-dimensional space inside the triangle, measured in square units (e.g., square meters). They are distinct concepts and are rarely the same value.
Forgetting Validation
It's easy to input three random numbers and assume they form a triangle. However, without checking the Triangle Inequality Theorem, the result may be meaningless. For example, you cannot have a triangle with sides 1, 2, and 10.
Angle Units
When using the SAS or ASA methods, ensure that the angles are provided in degrees, as that is the unit expected by the trigonometric formulas (Law of Sines and Law of Cosines). Using radians without conversion will lead to incorrect results.

Avoiding Errors

  • A triangle with sides 3, 4, 5 has a perimeter of 12 and an area of 6.
  • Using angles in radians? Convert to degrees first: degrees = radians * (180 / π).

Mathematical Derivations and Formulas

  • The Law of Cosines for the SAS case.
  • The Law of Sines for the ASA case.
  • The Distance Formula for the coordinate case.
Law of Cosines (for SAS)
When you have two sides (a, b) and the included angle (C), the third side (c) is found with: c = √(a² + b² - 2ab cos(C)). Once c is found, P = a + b + c.
Law of Sines (for ASA)
Given angle B, side a, and angle C, you first find angle A = 180° - B - C. Then, you find the other sides using the Law of Sines: b = (a sin(B)) / sin(A) and c = (a sin(C)) / sin(A). Finally, P = a + b + c.
Distance Formula (for Coordinates)
For two points (x₁, y₁) and (x₂, y₂), the distance between them is d = √((x₂-x₁)² + (y₂-y₁)²). This is applied three times for the three pairs of vertices (A-B, B-C, C-A) to find the lengths of the three sides. The perimeter is the sum of these three distances.

Formulas Used

  • SAS: a=10, b=12, C=50°. c = √(10² + 12² - 2*10*12*cos(50°)) ≈ 9.49. P ≈ 10 + 12 + 9.49 = 31.49.
  • ASA: a=8, B=45°, C=60°. A=75°. b = (8 * sin(45°))/sin(75°) ≈ 5.86. c = (8 * sin(60°))/sin(75°) ≈ 7.17. P ≈ 8 + 5.86 + 7.17 = 21.03.