Math
Law of sines
Use one side and its opposite angle with a second angle to determine the remaining triangle sides without the ambiguous side-side-angle case.
Calculators / Math / Law of cosines
Law of cosines calculator
Determine the third side of a triangle from two known sides and the angle between them, then calculate the enclosed area.
On this page
Opposite side5units
Area6units²
Calculator controls
How to calculate
Determine the third side of a triangle from two known sides and the angle between them, then calculate the enclosed area.
Opposite side = sqrt(a^2+b^2-2*a*b*cos(angle*pi/180)) units; Area = a*b*sin(angle*pi/180)/2 units²
Worked example
Side a: 3; Side b: 4; Included angle (degrees): 90.
Opposite side: 5 units; Area: 6 units².
Assumptions and limits
Determine the third side of a triangle from two known sides and the angle between them, then calculate the enclosed area. Use the same length unit throughout. Angles marked degrees use degrees. Results describe ideal geometric shapes.
Results are rounded for display; calculations use unrounded values. Read our calculation methodology.
Understanding the result
Determine the third side of a triangle from two known sides and the angle between them, then calculate the enclosed area.
When this tool is useful
Opposite side = sqrt(a^2+b^2-2*a*b*cos(angle*pi/180)) units; Area = a*b*sin(angle*pi/180)/2 units²
Understanding your inputs
| Input | What to enter |
|---|---|
| Side a | Enter a number of at least 1e-12 and no more than 1000000000000. |
| Side b | Enter a number of at least 1e-12 and no more than 1000000000000. |
| Included angle (degrees) | Enter a number of at least 0.000001 and no more than 179.999999. |
Frequently asked questions
What assumptions does this calculation use?
Determine the third side of a triangle from two known sides and the angle between them, then calculate the enclosed area. Use the same length unit throughout. Angles marked degrees use degrees. Results describe ideal geometric shapes.
How is the result calculated?
Opposite side = sqrt(a^2+b^2-2*a*b*cos(angle*pi/180)) units; Area = a*b*sin(angle*pi/180)/2 units²