Triangle Length Calculator
Enter any three known values of a triangle (sides and/or angles) and this calculator finds all the missing sides, angles, area, perimeter, altitudes, inradius, and circumradius. Choose your solving mode - SSS, SAS, ASA, AAS, or SSA - enter the values, and your complete triangle solution appears with a full step-by-step breakdown and a labeled diagram.
How to use this calculator
Select a solving mode from the dropdown: SSS if you know all three sides, SAS if you know two sides and the angle between them, ASA if you know two angles and the side between them, AAS if you know two angles and a side not between them, or SSA if you know two sides and an angle that is not between them. Enter your known values in the fields that appear, choose your preferred length unit, and all the missing sides, angles, area, perimeter, inradius, and circumradius are calculated immediately. The step-by-step panel shows every formula applied with your actual numbers substituted in.
Formulas used
The calculator uses three core rules of plane geometry. The Law of Sines states that a / sin(A) = b / sin(B) = c / sin(C), where lowercase letters are side lengths and uppercase letters are the angles opposite those sides. The Law of Cosines states that a^2 = b^2 + c^2 - 2bc cos(A), and similar forms for the other sides. These two laws cover every solving case except AAA, which is indeterminate. Area is computed as 0.5 * a * b * sin(C). The inradius r equals Area / s where s is the semiperimeter (P/2), and the circumradius R equals (a * b * c) / (4 * Area).
The SSA ambiguous case explained
When you know two sides and an angle that is not between them (SSA), the triangle may have zero, one, or two valid solutions. If the side opposite the given angle is shorter than the other given side and the given angle is acute, there may be two triangles that satisfy the conditions. The calculator returns the primary solution (the one with an acute angle B). The insight panel will flag when a second valid solution also exists, giving you the alternate angle B so you can solve the second triangle manually if needed.
What the outputs mean
Sides a, b, and c are the three edge lengths; side a is opposite angle A, b opposite B, and c opposite C. Angles A, B, and C are the interior angles at the three vertices; they always sum to 180 degrees. Area is the amount of flat surface enclosed. Perimeter is the total distance around the outside. The inradius r is the radius of the largest circle that fits inside the triangle (the inscribed circle). The circumradius R is the radius of the circle that passes through all three vertices (the circumscribed circle).
Triangle solving rules by known values
| Mode | Known values | Primary formula | Notes |
|---|---|---|---|
| SSS | 3 sides | Law of Cosines | Unique solution if triangle inequality holds |
| SAS | 2 sides + included angle | Law of Cosines | Unique solution |
| ASA | 2 angles + included side | Law of Sines | Unique solution; third angle = 180 - A - B |
| AAS | 2 angles + non-included side | Law of Sines | Unique solution; same as ASA after finding third angle |
| SSA | 2 sides + non-included angle | Law of Sines | Ambiguous: may give 0, 1, or 2 solutions |
| AAA | 3 angles only | None | Cannot determine side lengths without at least one side |
Choose your mode based on which three values you know.
Frequently asked questions
What information do I need to solve a triangle?
You need at least three pieces of information, and at least one of them must be a side length. Three angles alone (AAA) define the shape but not the size, so side lengths remain unknown. The five solvable cases are SSS (three sides), SAS (two sides, included angle), ASA (two angles, included side), AAS (two angles, non-included side), and SSA (two sides, non-included angle). This calculator handles all five cases.
Why does SSA sometimes give two answers?
In the SSA configuration, the given angle is not sandwiched between the two known sides. Depending on the relative lengths, you can swing the unknown side into two different positions that both satisfy the given angle, producing two distinct triangles. This is called the ambiguous case. If the side opposite the given angle is longer than the adjacent side, only one solution exists. If it is shorter and the angle is acute, two solutions are possible. If the math gives no valid triangle, there is no solution at all.
How does the calculator find angles from three sides (SSS)?
It applies the Law of Cosines three times. For angle A: A = arccos((b^2 + c^2 - a^2) / (2bc)). The same rearrangement finds B, and then C = 180 - A - B. The step-by-step panel shows the exact numbers for your inputs.
What is the difference between the inradius and the circumradius?
The inradius (r) is the radius of the inscribed circle, the largest circle that fits entirely inside the triangle, touching all three sides. The circumradius (R) is the radius of the circumscribed circle, the unique circle that passes through all three vertices. For any triangle, r = Area / s (where s is the semiperimeter) and R = (a * b * c) / (4 * Area).
Does it matter which side I label a, b, or c?
No, as long as you are consistent within a single calculation. The convention used here is that side a is opposite angle A, side b is opposite angle B, and side c is opposite angle C. Swapping labels will produce the same triangle geometry - just with different letters attached to each side and angle.
Can I use this for right triangles?
Yes. Enter the right angle (90 degrees) as one of the known angles and fill in the mode that matches what you know. For a right triangle with legs a and b and hypotenuse c, the Pythagorean theorem a^2 + b^2 = c^2 is a special case of the Law of Cosines with C = 90 degrees, so the results will be exact.