Calculate side C or angle C using the law of cosines from two sides and the included angle, with inch, ft, cm, m, degree, or radian units.
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Hinge Theorem Formula
This calculator finds the length of the third side of a triangle when you know two side lengths and the included angle between them. Strictly speaking, the hinge theorem is a comparison theorem, while the actual numeric side calculation is done with the closely related law of cosines. In practice, they describe the same geometric idea: as the included angle opens wider, the opposite side gets longer.
| Variable | Meaning | Requirement |
|---|---|---|
| a | First known side length | Must be positive |
| b | Second known side length | Must be positive |
| C | Included angle between side A and side B | Must be the angle formed by the two known sides |
| c | Unknown third side opposite angle C | Returned in the selected result unit |
How to Use the Calculator
- Select Side C or Included angle C, then enter the required lengths.
- Enter the length of side B.
- Enter the included angle between those two sides.
- Select degrees or radians so the angle is interpreted correctly.
- Calculate to find the selected unknown. Angle mode uses all three side lengths.
Important: the angle must be the one directly between the two known sides. If you use a different angle, the result will not represent the intended triangle.
What the Hinge Theorem Means
Imagine two triangles built from the same two side lengths. If one included angle is larger, the side across from that angle must also be larger. This is why the theorem is called the hinge theorem: the two known sides act like rigid arms connected by a hinge at the angle.
That comparison principle gives intuitive meaning to the calculator output. Keeping the two known sides fixed:
- a smaller included angle produces a shorter third side,
- a right angle produces a middle case,
- and a larger included angle produces a longer third side.
Angle Limits and Valid Input
For a real triangle, the included angle must be greater than zero and less than a straight angle.
If you are using radians, the same condition is:
Select the correct unit for each side. Mixed units are converted automatically before calculation. When solving angle C from three sides, the lengths must satisfy the strict triangle inequality: |aโb| < c < a+b.
Example
Suppose side A is 5, side B is 7, and the included angle is 60 degrees.
So the third side is approximately 6.245 units.
Useful Special Cases
When the included angle is 90 degrees, the formula becomes the Pythagorean relationship for a right triangle.
As the included angle gets very small, the third side approaches the difference of the two known sides. As the angle approaches 180 degrees, the third side approaches their sum.
Common Mistakes
- Using an angle that is not between the two known sides.
- Mixing degrees and radians.
- Selecting an incorrect unit for a side length.
- Using zero, negative lengths, or an angle of 0 or 180 degrees.
Where This Calculation Is Used
- triangle geometry and trigonometry problems,
- roof, frame, and brace layout,
- surveying and navigation,
- mechanical linkages and structural analysis,
- any situation where two sides and the included angle are known but the closing side is not.
If you know two sides and the angle between them, this calculator is the direct way to determine the missing third side accurately and quickly.
