Find Length In Triangle Calculator

Find missing side lengths of any triangle.

Calculate Side Length

Summary Table

Item	Value	Unit
Side a	–	units
Side b	–	units
Side c	–	units
Angle A	–	degrees
Angle B	–	degrees
Angle C	–	degrees
Area	–	sq. units

Summary of input and calculated values.

What is a Triangle Side Length Calculator?

A triangle side length calculator is a tool used to determine the length of an unknown side (or sides) of a triangle when you have sufficient information about its other sides and/or angles. Depending on the known values, different mathematical principles are applied, such as the Law of Cosines, the Law of Sines, or the Pythagorean theorem for right-angled triangles. This calculator helps students, engineers, architects, and anyone dealing with geometric problems to quickly find missing triangle dimensions.

It’s particularly useful when you can’t directly measure a side but know other properties of the triangle. For example, in surveying or astronomy, direct measurement is often impossible. A triangle side length calculator simplifies these calculations.

Common misconceptions include thinking any three values will define a triangle (e.g., three angles don’t define side lengths) or that all triangles are right-angled.

Triangle Side Length Formulas and Mathematical Explanation

The method used by a triangle side length calculator depends on the input data:

1. Law of Cosines (for SAS – Two Sides and Included Angle)

If you know two sides (a and b) and the angle C between them, you can find the third side (c) using the Law of Cosines:

c² = a² + b² - 2ab * cos(C)

From this, c = sqrt(a² + b² - 2ab * cos(C)). Similarly, you can find ‘a’ or ‘b’ if other sides and the included angle are known.

2. Law of Sines (for ASA/AAS – Two Angles and a Side)

If you know two angles (e.g., A and B) and one side (e.g., a), you first find the third angle (C = 180 – A – B), then use the Law of Sines:

a/sin(A) = b/sin(B) = c/sin(C)

To find side b: b = a * sin(B) / sin(A)
To find side c: c = a * sin(C) / sin(A)

3. Pythagorean Theorem (for Right-Angled Triangles)

If the triangle is right-angled (one angle is 90°), and you know two sides, you can find the third:

• If you know legs a and b: Hypotenuse c = sqrt(a² + b²)
• If you know leg a and hypotenuse c: Leg b = sqrt(c² - a²)
• If you know leg b and hypotenuse c: Leg a = sqrt(c² - b²)

Variables Table

Variable	Meaning	Unit	Typical Range
a, b, c	Lengths of the sides of the triangle	units (e.g., m, cm, ft)	> 0
A, B, C	Angles opposite sides a, b, c respectively	degrees (or radians)	> 0 and < 180 (A+B+C=180)
cos(C), sin(A), sin(B), sin(C)	Trigonometric functions of the angles	dimensionless	-1 to 1

Practical Examples

Example 1: Using Law of Cosines (SAS)

Suppose you have a triangle with side a = 8m, side b = 10m, and the angle C between them is 60 degrees. To find side c:

c = sqrt(8² + 10² - 2 * 8 * 10 * cos(60°))
c = sqrt(64 + 100 - 160 * 0.5)
c = sqrt(164 - 80) = sqrt(84) ≈ 9.17 m

Our triangle side length calculator would give you this result instantly.

Example 2: Using Law of Sines (AAS)

Imagine you know side a = 12 cm, angle A = 40°, and angle B = 70°. First, find angle C = 180 – 40 – 70 = 70°. Then find side b:

b = 12 * sin(70°) / sin(40°)
b ≈ 12 * 0.9397 / 0.6428 ≈ 17.55 cm

The triangle side length calculator handles these steps smoothly.

Example 3: Using Pythagorean Theorem

A right triangle has legs of 3 ft and 4 ft. The hypotenuse c is:

c = sqrt(3² + 4²) = sqrt(9 + 16) = sqrt(25) = 5 ft

How to Use This Triangle Side Length Calculator

1. Select the Method: Choose what you know from the “What do you know?” dropdown (SAS, ASA/AAS, Right Triangle – Two Legs, Right Triangle – Leg and Hypotenuse).
2. Enter Known Values: Input the lengths of the known sides and/or the measures of the known angles (in degrees) into the corresponding fields that appear. Ensure the units are consistent.
3. Calculate: Click the “Calculate” button or just change the input values.
4. View Results: The calculator will display the length(s) of the unknown side(s) as the primary result, along with intermediate values like other angles or area where applicable, and the formula used.
5. Visualize: The SVG chart will attempt to draw the triangle based on the inputs or calculated values.
6. Check Summary: The table summarizes all side lengths, angles, and area.

The triangle side length calculator provides quick and accurate results based on standard geometric formulas.

Key Factors That Affect Results

• Accuracy of Input Values: Small errors in measuring sides or angles can lead to larger errors in calculated lengths, especially with the Law of Sines and Cosines.
• Units: Ensure all side lengths are in the same units. The result will be in those same units. Angles are assumed to be in degrees.
• Angle Sum: For ASA/AAS, the sum of the two input angles must be less than 180 degrees to form a valid triangle. The calculator checks for this.
• Triangle Inequality Theorem: For any triangle, the sum of the lengths of any two sides must be greater than the length of the third side. While calculating a side, this isn’t directly an input factor but a condition for a valid triangle formed by the results.
• Right Angle Assumption: When using the Pythagorean theorem options, you are assuming one angle is exactly 90 degrees.
• Rounding: The number of decimal places used in calculations and results can affect precision. Our triangle side length calculator uses sufficient precision internally.

Frequently Asked Questions (FAQ)

What if I know three sides (SSS)?

If you know three sides, you can find the angles using the Law of Cosines, but you don’t need a triangle side length calculator as all side lengths are already known. You might need a triangle angle calculator.

Can I use this calculator for any triangle?

Yes, as long as you have the required information for one of the supported methods (SAS, ASA/AAS, or right-triangle properties).

What units should I use?

You can use any unit of length (cm, meters, inches, feet, etc.) as long as you are consistent for all input sides. The output side length will be in the same unit.

What if my two input angles for ASA/AAS add up to 180 or more?

The calculator will indicate an error because a triangle cannot have two angles summing to 180 degrees or more.

How does the triangle side length calculator find other angles in SAS?

After finding the third side ‘c’ using the Law of Cosines, it can use the Law of Sines or Law of Cosines again to find the remaining angles (e.g., `a/sin(A) = c/sin(C)`).

Is the triangle drawing always to scale?

The SVG drawing attempts to represent the triangle, but due to scaling constraints to fit the box, it may not be perfectly to scale, especially for triangles with very different side lengths or extreme angles. It’s illustrative.

Can I find the area using this calculator?

Yes, the calculator also computes and displays the area of the triangle using appropriate formulas based on the inputs (e.g., 0.5 * a * b * sin(C) for SAS).

What if I enter negative values for sides or angles?

The calculator will show an error, as side lengths and angles (in the context of triangle geometry) must be positive, and angles less than 180.

Related Tools and Internal Resources

• Area of Triangle Calculator: Calculate the area given various inputs.
• Right Triangle Calculator: A specialized calculator for right-angled triangles.
• Law of Sines Calculator: Focuses specifically on the Law of Sines.
• Law of Cosines Calculator: Focuses specifically on the Law of Cosines.
• Triangle Angle Calculator: Find missing angles if you know sides or other angles.
• Geometry Calculators: A collection of various geometry-related calculators.