Free · Triangle geometry
Triangle Perimeter from Sides or Angles
Find a triangle boundary length using three sides, two sides with their included angle, or a side and two adjacent angles.
- Formula & worked example
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Calculator inputs
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How to use this calculator
- Enter the known values in the units shown. Results update as you type.
- Where results are editable, change one to solve backwards. Lock a value to hold it fixed.
- Use the worked example to check the method. Reset restores the starting fields.
Private by default
Inputs and results stay in this browser tab. Bookify does not upload or store the values you enter.
Formula and method
A triangle with sides 3, 4 and 5 has perimeter 12 in the same length unit.
P = a+b+c; c² = a²+b²−2ab cos γ; a/sin α = b/sin β = c/sin γ; α+β+γ = π.
Worked example
Enter these known values and leave the other values blank.
- a
- 3 cm
- b
- 4 cm
- c
- 5 cm
- Perimeter
- 12 cm
Assumptions and limitations
- All lengths must be positive and use a common unit basis. A nondegenerate triangle has all three strict triangle inequalities.
- Select three sides, two sides with their included angle, or one side and its two adjacent angles. All angles must be positive and their sum is exactly pi radians or 180 degrees.
- Near-degenerate triangles can exceed finite numerical resolution: a rounded perimeter at a strict boundary triggers a warning even when the ideal triangle is nondegenerate. Use greater precision or less extreme geometry.
- In two-angle mode, the two visible adjacent angles establish the third angle by the angle sum; an entered side and perimeter must agree within a relative residual of 1e-40.
- The two-side perimeter must lie strictly between twice the larger known side and twice the sum of both sides. Clear an unknown before a supported inverse. Ambiguous angle-only reconstruction may require another independent value.
Common questions
Can I use two sides and an angle?
Yes. Choose the included angle between the known sides. The law of cosines supplies the third side, then all three are added.
Which two angles go with the known side?
In the two-angle mode, the known side lies between the two entered angles. Its opposite angle equals 180 degrees minus their sum.
References
- NIST DLMF: trigonometric identities
- OpenStax: law of sines
- OpenStax: law of cosines
- Calculation definition and unit reference
Replace the captured 3.1415-radian triangle angle sum with mathematical pi in every angle-sum inverse. Use equivalent half-angle identities for the two-side third length and included-angle inverse to avoid subtracting nearly equal squared sides. Establish all ASA angles by the angle sum, eliminating principal-arcsine precedence; calculate ASA side and perimeter from the two visible adjacent angles. Use the exact pi angle boundary, strict positive triangle sides and angle margins, and enforce the perimeter interval required by a nondegenerate two-side triangle inverse. Reject incompatible overdetermined ASA inputs with an explicit relative residual bound of 1e-40 using only visible fields.