Find The Triangle Calculator

Enter the lengths of the three sides of your triangle to find its area, perimeter, angles, and type using our triangle calculator.

Property	Value
Side a	3
Side b	4
Side c	5
Area	–
Perimeter	–
Type	–
Angle A	–
Angle B	–
Angle C	–

Summary of triangle properties based on input sides.

What is a Triangle Calculator?

A triangle calculator is a specialized tool designed to determine various properties of a triangle based on a given set of inputs. Most commonly, a triangle calculator will take the lengths of the three sides (a, b, and c) and calculate the triangle's area, perimeter, the angles opposite each side, and classify the type of triangle (e.g., equilateral, isosceles, scalene, right-angled, acute, or obtuse). This particular triangle calculator focuses on the Side-Side-Side (SSS) case.

Anyone studying geometry, trigonometry, or needing to solve practical problems involving triangles, such as engineers, architects, students, and DIY enthusiasts, can benefit from using a triangle calculator. It saves time and reduces the chance of manual calculation errors.

A common misconception is that any three lengths can form a triangle. However, the Triangle Inequality Theorem must be satisfied: the sum of the lengths of any two sides of a triangle must be greater than the length of the third side. Our triangle calculator checks this condition.

Triangle Calculator Formula and Mathematical Explanation

When given three sides (a, b, c), our triangle calculator uses the following formulas:

1. Triangle Inequality Check: It first verifies if a + b > c, a + c > b, and b + c > a. If not, the sides do not form a valid triangle.
2. Perimeter (P): P = a + b + c
3. Semi-perimeter (s): s = P / 2 = (a + b + c) / 2
4. Area (A) using Heron's Formula: A = √(s * (s – a) * (s – b) * (s – c))
5. Angles (A, B, C) using the Law of Cosines:
   • cos(A) = (b² + c² – a²) / (2bc) ⇒ A = arccos((b² + c² – a²) / (2bc)) * (180/π)
   • cos(B) = (a² + c² – b²) / (2ac) ⇒ B = arccos((a² + c² – b²) / (2ac)) * (180/π)
   • cos(C) = (a² + b² – c²) / (2ab) ⇒ C = arccos((a² + b² – c²) / (2ab)) * (180/π)
6. Type of Triangle:
   • If a = b = c, it's Equilateral.
   • If a = b or b = c or a = c (but not all three), it's Isosceles.
   • If a ≠ b ≠ c, it's Scalene.
   • Based on angles (after calculating them or using squares of sides): If one angle is 90° (e.g., a² + b² = c²), it's Right-angled. If one angle > 90° (e.g., c² > a² + b² where c is longest), it's Obtuse. If all angles < 90°, it's Acute.

Variables Used in the Triangle Calculator

Variable	Meaning	Unit	Typical Range
a, b, c	Lengths of the sides	(e.g., cm, m, inches)	> 0
P	Perimeter	Same as sides	> 0
s	Semi-perimeter	Same as sides	> 0
A	Area	(e.g., cm², m², inches²)	> 0
A, B, C	Angles opposite sides a, b, c	Degrees (°)	0° to 180°

Practical Examples (Real-World Use Cases)

Example 1: The Classic 3-4-5 Triangle

Suppose you have a piece of land with sides 30 meters, 40 meters, and 50 meters.

• Input: Side a = 30, Side b = 40, Side c = 50
• Perimeter = 30 + 40 + 50 = 120 meters
• Semi-perimeter (s) = 120 / 2 = 60
• Area = √(60 * (60-30) * (60-40) * (60-50)) = √(60 * 30 * 20 * 10) = √360000 = 600 square meters
• Angles: Using the Law of Cosines, you'd find angles of 36.87°, 53.13°, and 90°.
• Type: Right-angled Scalene Triangle
• Our triangle calculator quickly gives these results.

Example 2: An Isosceles Triangle

Imagine a roof gable with two sides of 5 meters and a base of 8 meters.

• Input: Side a = 5, Side b = 5, Side c = 8
• Perimeter = 5 + 5 + 8 = 18 meters
• Semi-perimeter (s) = 18 / 2 = 9
• Area = √(9 * (9-5) * (9-5) * (9-8)) = √(9 * 4 * 4 * 1) = √144 = 12 square meters
• Angles: Approx. 36.87°, 36.87°, and 106.26°
• Type: Obtuse Isosceles Triangle
• The triangle calculator confirms this.

How to Use This Triangle Calculator

1. Enter Side Lengths: Input the lengths of the three sides (a, b, and c) into the respective fields. Ensure they are positive numbers.
2. View Results: The triangle calculator automatically updates and displays the Area (primary result), Perimeter, Type, and Angles A, B, and C as you type.
3. Check Validity: If the sides entered cannot form a triangle, an error message will appear.
4. Read the Table: The table summarizes all input and calculated values.
5. Examine the Chart: The bar chart visualizes the lengths of the sides.
6. Use the Buttons:
   • "Calculate": Manually triggers calculation (though it's automatic on input).
   • "Reset": Clears inputs and results to default values.
   • "Copy Results": Copies the main results and inputs to your clipboard.

Use the results from the triangle calculator to understand the geometry of your triangle, whether for academic purposes or practical applications like construction or land measurement. You might also be interested in our Area Calculator for other shapes.

Key Factors That Affect Triangle Calculator Results

1. Side Lengths (a, b, c): The fundamental inputs. Changing any side length directly impacts perimeter, area, angles, and type. The triangle calculator relies entirely on these.
2. Triangle Inequality Theorem: The entered side lengths must satisfy a+b>c, a+c>b, and b+c>a. If not, no triangle exists, and the triangle calculator will indicate this.
3. Relative Side Lengths: The ratios between side lengths determine the angles and the type of triangle (equilateral, isosceles, scalene, right, acute, obtuse).
4. Longest Side vs. Sum of Squares of Others: Comparing c² to a² + b² (where c is the longest side) determines if the triangle is right (equal), obtuse (c² >), or acute (c² <). Our triangle calculator performs this check.
5. Units of Measurement: Ensure all side lengths are in the same units. The area will be in square units of that measurement, and the perimeter in those units. The triangle calculator assumes consistent units.
6. Precision of Input: The more decimal places you use for side lengths, the more precise the calculated area and angles will be, though rounding is inherent in angle calculations involving arccos.

Frequently Asked Questions (FAQ)

1. What is a triangle calculator used for?

A triangle calculator is used to find the area, perimeter, angles, and type of a triangle based on given inputs, usually the lengths of the three sides (SSS), or other combinations like SAS or ASA in more advanced calculators.

2. What is Heron's formula?

Heron's formula is used by this triangle calculator to find the area of a triangle given the lengths of its three sides. Area = √(s(s-a)(s-b)(s-c)), where s is the semi-perimeter.

3. How do you know if three sides form a valid triangle?

The sum of the lengths of any two sides must be greater than the length of the third side. Our triangle calculator checks this.

4. How does the triangle calculator find the angles?

It uses the Law of Cosines, which relates the lengths of the sides of a triangle to the cosine of one of its angles.

5. Can this triangle calculator solve triangles with angles given?

This specific triangle calculator is designed for the SSS (Side-Side-Side) case. You would need a different tool or mode for ASA or SAS cases, though you can explore our Angle Calculator for related concepts.

6. What types of triangles can this calculator identify?

It can identify Equilateral, Isosceles, Scalene, Right-angled, Obtuse, and Acute triangles based on the side lengths and calculated angles.

7. What units should I use for the sides?

You can use any consistent units (cm, m, inches, feet, etc.). The area will be in the square of those units, and the perimeter will be in those units. The triangle calculator doesn't convert units.

8. Is it possible to get an area of zero?

If the three points are collinear (form a straight line), the area would be zero, but this would also violate the Triangle Inequality Theorem for a non-degenerate triangle. The triangle calculator will likely flag it as invalid sides.

Related Tools and Internal Resources

• Area Calculator: Calculate the area of various shapes, including triangles using different formulas.
• Perimeter Calculator: Find the perimeter of different geometric figures.
• Geometry Formulas: A collection of common geometry formulas, including those used by the triangle calculator.
• Math Tools: A suite of online math calculators and solvers.
• Right Triangle Calculator: A specialized calculator for right-angled triangles, exploring Pythagoras' theorem and trigonometric functions.
• Angle Calculator: Tools for angle conversions and calculations in geometry.