Find The Missing Side Of Triangle Calculator

Use this calculator to find the missing side of a triangle given other sides and/or angles. Select the case that matches your known values.

Visual representation of the triangle (not to scale).

What is a Missing Side of a Triangle Calculator?

A missing side of a triangle calculator is a tool used to determine the length of an unknown side of a triangle when you have sufficient information about its other sides and/or angles. Triangles are fundamental geometric shapes, and finding their side lengths is crucial in various fields like engineering, architecture, physics, and navigation. This calculator can handle different scenarios, including right-angled triangles and non-right-angled triangles (oblique triangles) using the Law of Sines and the Law of Cosines.

Who should use it? Students learning trigonometry, engineers, architects, surveyors, or anyone needing to solve for triangle dimensions. The missing side of a triangle calculator simplifies complex calculations, saving time and reducing errors.

Common misconceptions include thinking all triangles can be solved with just any two pieces of information. You typically need three pieces of information (like two sides and an angle, or two angles and a side, or three sides – though if you know three sides, none are missing!), and the type of information determines the method used (Pythagorean theorem, Law of Sines, or Law of Cosines).

Missing Side of a Triangle Calculator: Formulas and Mathematical Explanation

To find the missing side, different formulas are used depending on the type of triangle and the known information:

1. Right-Angled Triangle (Pythagorean Theorem)

If the triangle is right-angled (one angle is 90°), and you know two sides, you can use the Pythagorean theorem: a² + b² = c², where 'c' is the hypotenuse (the side opposite the right angle), and 'a' and 'b' are the other two sides (legs).

• If 'a' and 'b' are known, c = √(a² + b²)
• If 'a' and 'c' are known, b = √(c² - a²)
• If 'b' and 'c' are known, a = √(c² - b²)

2. Oblique Triangle – Law of Cosines (SAS case)

If you know two sides and the included angle (the angle between those two sides – SAS), you can use the Law of Cosines to find the third side. If you know sides 'a', 'b', and the angle 'C' between them, the missing side 'c' is found by: c² = a² + b² - 2ab * cos(C). So, c = √(a² + b² - 2ab * cos(C)). Remember to convert angle C to radians for the cos function if your calculator/language requires it (JavaScript's Math.cos expects radians).

3. Oblique Triangle – Law of Sines (AAS or ASA case)

If you know two angles and one side (AAS or ASA), you can use the Law of Sines. First, find the third angle (since A + B + C = 180°). Then, the Law of Sines states: a/sin(A) = b/sin(B) = c/sin(C). If you know angles A, B, and side 'a', you can find side 'b': b = (a * sin(B)) / sin(A).

Variables Table:

Variables used in triangle calculations.

Variable	Meaning	Unit	Typical Range
a, b, c	Lengths of the sides of the triangle	Units of length (e.g., cm, m, inches)	Positive numbers
A, B, C	Angles opposite sides a, b, c respectively	Degrees (or radians in calculations)	0° – 180° (sum is 180°)
√	Square root	N/A	N/A
cos, sin	Cosine and Sine trigonometric functions	N/A	N/A

Practical Examples (Real-World Use Cases)

Example 1: Right-Angled Triangle

A carpenter is building a ramp. The base of the ramp (side 'a') is 12 feet long, and the height (side 'b') is 5 feet. What is the length of the ramp surface (hypotenuse 'c')?

• Known: a = 12, b = 5 (and it's a right angle)
• Formula: c = √(a² + b²) = √(12² + 5²) = √(144 + 25) = √169 = 13
• Result: The ramp surface is 13 feet long. Our missing side of a triangle calculator would confirm this.

Example 2: SAS Case (Law of Cosines)

Two ships leave a port at the same time. Ship A travels at 10 knots on a bearing, and Ship B travels at 12 knots on another bearing, with an angle of 45° between their paths. How far apart are the ships after one hour?

• Known: side a = 10 nautical miles, side b = 12 nautical miles, Angle C = 45°
• Formula: c = √(a² + b² – 2ab * cos(C)) = √(10² + 12² – 2 * 10 * 12 * cos(45°)) = √(100 + 144 – 240 * 0.7071) = √(244 – 169.7) = √74.3 ≈ 8.62 nautical miles.
• Result: The ships are approximately 8.62 nautical miles apart after one hour. You can verify this with the missing side of a triangle calculator.

How to Use This Missing Side of a Triangle Calculator

1. Select the Case: Choose the option from the dropdown that matches the information you have (e.g., "Right-angled: know sides a & b", "SAS", "AAS/ASA").
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.
3. View Results: The calculator will automatically display the length of the missing side in the "Primary Result" section as you type. It will also show other calculated values like angles (if applicable) and the formula used.
4. Reset: Use the "Reset" button to clear inputs and start over.
5. Copy: Use the "Copy Results" button to copy the findings to your clipboard.

The calculator provides immediate feedback, making it easy to see how changing one value affects the missing side and other triangle properties. For more complex problems, consider our geometry calculators.

Key Factors That Affect Missing Side Results

• Type of Triangle: Whether it's right-angled or oblique significantly changes the formula used (Pythagorean vs. Sines/Cosines). Our missing side of a triangle calculator handles these.
• Known Sides: The lengths of the known sides directly influence the possible length of the missing side.
• Known Angles: The angles determine the proportions of the triangle and are crucial for the Law of Sines and Cosines.
• Included Angle (SAS): In the SAS case, the angle between the two known sides is vital. A larger angle generally means a longer third side.
• Angle Sum: The sum of angles in any triangle must be 180°. This is used in the AAS/ASA case to find the third angle before applying the Law of Sines.
• Units: Ensure all side lengths are in the same units. The result will be in the same unit. Check out our triangle area calculator for related calculations.

Frequently Asked Questions (FAQ)

Q: Can I find a missing side if I only know the angles? A: No. Knowing only the angles defines the shape (similarity) but not the size of the triangle. You need at least one side length.

Q: What is the hypotenuse? A: In a right-angled triangle, the hypotenuse is the longest side, opposite the right angle (90°). Our missing side of a triangle calculator identifies this when 'c' is calculated or given in right-angled cases.

Q: What if the sum of angles A and B in AAS/ASA is more than 180°? A: That's not possible for a valid triangle. The calculator will likely show an error or an invalid result. The sum of all three angles must be exactly 180°.

Q: Can I use this calculator for any triangle? A: Yes, it covers right-angled triangles and oblique triangles using the SAS and AAS/ASA input methods. For SSS (three sides known), no side is missing, but you could use the Law of Cosines to find angles. We also have a dedicated right-triangle calculator.

Q: Why does the SAS case need the *included* angle? A: The Law of Cosines formula specifically relates the third side to the other two sides and the angle *between* them.

Q: What are the Law of Sines and Law of Cosines? A: They are trigonometric laws relating the sides and angles of *any* triangle (not just right-angled). The Law of Sines relates sides to the sines of opposite angles, and the Law of Cosines relates the square of a side to the squares of the other two and the cosine of the included angle.

Q: What if my inputs result in an impossible triangle (e.g., in SAS, a + b < c)? A: For the SAS and AAS/ASA cases, as long as angles are valid, a triangle is usually possible. For right-angled triangles, if c² – a² is negative, it's impossible. The calculator should handle these edge cases.

Q: How accurate is this missing side of a triangle calculator? A: The calculations are based on standard trigonometric formulas and are as accurate as the input values and the precision of the JavaScript Math functions.

Related Tools and Internal Resources

• Triangle Area Calculator: Calculate the area of a triangle given various inputs.
• Right-Triangle Calculator: A specialized calculator for right-angled triangles.
• Law of Sines Calculator: Solve triangles using the Law of Sines.
• Law of Cosines Calculator: Solve triangles using the Law of Cosines.
• Pythagorean Theorem Calculator: Specifically for finding sides in right-angled triangles.
• Geometry Calculators: A collection of various geometry-related calculators.