Find Other Side Of Triangle Calculator

Easily calculate the missing side of a right-angled triangle or any triangle (given SAS) using our Triangle Side Calculator. Input the known values to find the unknown side instantly.

1. Right-Angled Triangle Side Calculator (Pythagorean Theorem)

Enter any two sides of a right-angled triangle to find the third. Leave the field for the side you want to calculate blank.

Formula Used: Pythagorean Theorem (a² + b² = c²), rearranged based on the unknown side.

2. Any Triangle Side Calculator (SAS – Law of Cosines)

Enter two sides (b and c) and the included angle (A) to find the third side (a).

Formula Used: Law of Cosines (a² = b² + c² - 2bc * cos(A)).

Results Visualization

Side	Length
a	—
b	—
c	—
Angle A (deg)	—

Table showing the lengths of the sides and angle used/calculated. “a, b, c” refer to sides from the active calculator (right-angled or SAS).

Bar chart comparing the lengths of the triangle sides calculated.

What is a Triangle Side Calculator?

A Triangle Side Calculator is a tool used to find the length of an unknown side of a triangle when enough other information (sides and/or angles) is provided. Our calculator helps you find other side of triangle using either the Pythagorean theorem for right-angled triangles or the Law of Cosines for triangles where two sides and the included angle (SAS) are known. It’s a handy tool for students, engineers, architects, and anyone working with geometric figures.

You can use a find other side of triangle calculator to solve for the hypotenuse or a leg in a right triangle, or the third side in any triangle if you have SAS data.

Who Should Use It?

• Students: Learning trigonometry and geometry can use it to check their work or understand concepts.
• Engineers and Architects: For calculating dimensions in designs and structures.
• Builders and Carpenters: To determine lengths for construction projects.
• DIY Enthusiasts: For various home projects involving angles and lengths.

Common Misconceptions

One common misconception is that you can find a side with any two pieces of information. You need specific combinations: two sides of a right triangle, or two sides and the *included* angle for a non-right triangle (SAS), or two angles and a side (ASA or AAS, for which you’d use the Law of Sines, not directly covered by the SAS part of this calculator but related).

Triangle Side Formulas and Mathematical Explanation

The methods used by this Triangle Side Calculator depend on the type of triangle and the information given.

1. Right-Angled Triangle (Pythagorean Theorem)

For a right-angled triangle with sides ‘a’ and ‘b’ forming the right angle, and ‘c’ being the hypotenuse (the side opposite the right angle), the Pythagorean theorem states:

a² + b² = c²

From this, we can find any side if the other two are known:

• To find hypotenuse c: c = √(a² + b²)
• To find side a: a = √(c² - b²) (c must be > b)
• To find side b: b = √(c² - a²) (c must be > a)

2. Any Triangle – SAS (Law of Cosines)

When you know two sides (say ‘b’ and ‘c’) and the angle between them (angle A), you can find the third side (‘a’) using the Law of Cosines:

a² = b² + c² - 2bc * cos(A)

So, side ‘a’ is: a = √(b² + c² - 2bc * cos(A))

Similarly, if you wanted to find side ‘b’ given ‘a’, ‘c’, and angle B: b² = a² + c² - 2ac * cos(B), and for side ‘c’ given ‘a’, ‘b’, and angle C: c² = a² + b² - 2ab * cos(C).

Variables Table

Variable	Meaning	Unit	Typical Range
a, b	Lengths of the legs in a right-angled triangle, or sides of any triangle	Length (e.g., cm, m, inches)	> 0
c	Length of the hypotenuse in a right-angled triangle, or a side of any triangle	Length (e.g., cm, m, inches)	> 0
A, B, C	Angles of the triangle (in degrees)	Degrees	0° – 180° (sum = 180°)
cos(A)	Cosine of angle A	Dimensionless	-1 to 1

Practical Examples (Real-World Use Cases)

Example 1: Right-Angled Triangle – Building a Ramp

Suppose you are building a ramp that needs to cover a horizontal distance (side b) of 12 feet and reach a height (side a) of 3 feet. You want to find the length of the ramp surface (hypotenuse c).

• Side a = 3 feet
• Side b = 12 feet
• Hypotenuse c = ?

Using the Pythagorean theorem: c = √(3² + 12²) = √(9 + 144) = √153 ≈ 12.37 feet. The ramp surface needs to be about 12.37 feet long.

You would enter 3 for ‘Side a’, 12 for ‘Side b’, and leave ‘Hypotenuse c’ blank in the first calculator.

Example 2: Any Triangle (SAS) – Land Surveying

A surveyor measures two sides of a triangular piece of land as 150 meters (side b) and 200 meters (side c). The angle between these two sides (Angle A) is measured as 50 degrees. They want to find the length of the third side (a).

• Side b = 150 m
• Side c = 200 m
• Angle A = 50 degrees
• Side a = ?

Using the Law of Cosines: a² = 150² + 200² – 2 * 150 * 200 * cos(50°)
a² = 22500 + 40000 – 60000 * 0.6428 ≈ 62500 – 38568 = 23932
a ≈ √23932 ≈ 154.7 meters. The third side is about 154.7 meters long.

You would use the second calculator, entering 150 for ‘Side b’, 200 for ‘Side c’, and 50 for ‘Angle A’.

How to Use This Triangle Side Calculator

This page has two calculators:

1. Right-Angled Triangle Side Calculator:
   • Identify the two sides you know (a, b, or c).
   • Enter their lengths into the corresponding input fields (“Side a”, “Side b”, “Hypotenuse c”).
   • Leave the field for the side you want to find blank.
   • Click “Calculate”. The missing side’s length will be displayed, along with intermediate steps.
2. Any Triangle Side Calculator (SAS):
   • Use this when you know two sides and the angle *between* them (Side-Angle-Side).
   • Enter the lengths of the two known sides (‘b’ and ‘c’) and the measure of the included angle (‘A’ in degrees).
   • Click “Calculate Side a”. The length of the side opposite angle A (‘a’) will be shown.

Reading the Results

The main result is shown prominently. Intermediate values and the formula used are also displayed to help you understand the calculation. The table and chart visualize the side lengths.

Key Factors That Affect Triangle Side Calculation Results

1. Known Values: The accuracy and type of known values (sides, angles) directly determine the result and the method used. For the Pythagorean theorem calculator part, two sides are needed.
2. Triangle Type: Whether it’s a right-angled triangle or not dictates if Pythagoras or the Law of Cosines/Sines is applicable. Our first tool is for right-angled triangles, the second (SAS) for any triangle using the Law of Cosines calculator logic.
3. Included Angle (SAS): For the Law of Cosines, the angle must be the one *between* the two known sides.
4. Units: Ensure all side lengths are in the same units (e.g., all meters or all inches). The result will be in the same unit.
5. Angle Units: Angles must be entered in degrees for the SAS calculator.
6. Accuracy of Input: Small errors in input values, especially angles in the Law of Cosines, can lead to noticeable differences in the calculated side length.

Frequently Asked Questions (FAQ)

Q1: What if I only know the angles of a triangle? Can I find the sides?

A1: No, if you only know the angles (AAA), you know the shape of the triangle but not its size. There are infinitely many triangles with the same angles but different side lengths (similar triangles). You need at least one side length to determine the others using the Law of Sines calculator.

Q2: Can I use this calculator for any triangle?

A2: The first part is specifically for right-angled triangles. The second part (SAS) is for any triangle where you know two sides and the included angle. If you know other combinations (like ASA, AAS, SSS for finding angles), you’d need the Law of Sines or Cosines applied differently, or our triangle angle calculator.

Q3: What units should I use for the sides?

A3: You can use any unit of length (cm, meters, inches, feet, etc.), but be consistent. If you input sides in cm, the result will be in cm.

Q4: How accurate are the results from this find other side of triangle calculator?

A4: The calculations are based on standard mathematical formulas and are as accurate as the input you provide. The results are rounded to four decimal places.

Q5: What is the hypotenuse?

A5: In a right-angled triangle, the hypotenuse is the longest side, opposite the right angle (90 degrees).

Q6: What if my inputs result in an error or an impossible triangle?

A6: The calculator has basic validation. For example, in a right triangle, the hypotenuse must be longer than either leg. For SAS, the angle must be between 0 and 180 degrees. If inputs don’t form a valid triangle or scenario, an error or note will be shown.

Q7: Can I calculate angles with this tool?

A7: No, this Triangle Side Calculator is designed to find side lengths. To find angles, you would typically use the Law of Cosines (if you know all three sides – SSS) or the Law of Sines (if you have other combinations), or a dedicated triangle angle calculator.

Q8: Does the SAS calculator work for right-angled triangles too?

A8: Yes, if you use 90 degrees as the included angle in the Law of Cosines, it simplifies to the Pythagorean theorem (since cos(90°)=0). However, it’s easier to use the dedicated right-angled calculator if you know it’s a right triangle.