Triangle Side Calculator
Calculate the third side, perimeter, and area based on two sides and their included angle. Triangle Side Calculator helps students, engineers, architects, and surveyors solve triangle measurements efficiently.
Highlights
- Calculate a triangle’s missing third side using two sides and their included angle.
- Get accurate triangle perimeter and area results instantly.
- Uses the Law of Cosines and sine-based area calculation.
- Supports degrees and radians for angle inputs.
- Works with multiple length units, including metric and imperial measurements.
Table of Contents
How Triangle Side Calculator Works?
The Triangle Side Calculator uses two known side lengths, a and b, and the angle between them, C, to determine the third side, perimeter, and area of a triangle. The included angle must be between 0.0001 and 179.9999 degrees or radians, and both side values must be positive.
The missing side is calculated with the Law of Cosines. This generalizes the Pythagorean theorem for triangles that are not necessarily right-angled:
Taking the positive square root gives the third side:
The perimeter is the sum of the three side lengths:
The area is found from two sides and their included angle using the sine formula:
For example, with Side A = 5, Side B = 7, and an included angle of 60 degrees, the calculator gives a third side of approximately 6.245, a perimeter of approximately 18.245, and an area of approximately 15.155 square units.
| Output | Calculation |
|---|---|
| Third Side | c = √(a2 + b2 − 2ab cos(C)) |
| Perimeter | P = a + b + c |
| Area | A = ½ab sin(C) |
Practical Use Cases
Students can use the calculator to check geometry and trigonometry exercises involving two sides and a non-right included angle, then compare the calculated third side, perimeter, and area with their own work.
Engineers can estimate the length of an unknown member or separation between two measured points when two lengths and the angle between them are known. The perimeter and area results can also support preliminary design calculations.
Architects can evaluate triangular floor plans, façade sections, roof layouts, and other irregular triangular features from selected side measurements and an included angle.
Surveyors can determine the distance between two endpoints from two measured distances and their included angle, while also obtaining the boundary length and enclosed triangular area for site assessments.
Accuracy and Limitations
The calculator applies the Law of Cosines and the included-angle area formula using the entered decimal values, with results subject to normal floating-point arithmetic and displayed rounding. Side A and Side B must each be between 0.0001 and 1,000,000, the angle must be strictly between 0 and 180 degrees or radians, and both side values must use the same selected length unit. Extremely small angles, angles very close to 180 degrees, or nearly equal side lengths can make the calculated third side sensitive to small measurement or rounding errors. The calculator does not verify real-world measurement quality or account for surveying, construction, or instrument uncertainty.
Quick Examples
Example #1: Standard Triangle
Calculate the third side, perimeter, and area for a triangle with two meter-based sides and a 60-degree included angle.
Example #2: Small Engineering Triangle
Determine triangle measurements using centimeter-based side lengths and a 45-degree included angle.
Example #3: Surveying Triangle
Calculate triangle dimensions from two feet-based sides and an included angle provided in radians.
