Solving triangle...
Choose SSS, SAS, ASA, or AAS and enter three known values to solve the triangle.
What Is a Triangle Calculator?
A triangle calculator finds missing sides, angles, area, and other properties when enough measurements are known. This solver covers SSS, SAS, ASA, and AAS, the four common patterns that determine one unique triangle without an ambiguous second solution.
Choose the pattern that matches the known values, keep the standard labels consistent, and enter any positive decimal side lengths or angles in degrees. Results include full working, classification, heights, radii, and a diagram whose proportions come from the solved side lengths rather than a generic triangle shape.
Which Triangle Formula Solves Each Case?
Highlighted marks represent the known measurements. SSS and SAS begin with the Law of Cosines, while ASA and AAS first use the 180° angle sum and then the Law of Sines.
SSS
All three sides
Law of Cosines
SAS
Two sides + included angle
Law of Cosines
ASA
Two angles + included side
Angle Sum + Law of Sines
AAS
Two angles + non-included side
Angle Sum + Law of Sines
Triangle Side and Angle Naming
Capital letters name angles and vertices; lowercase letters name the opposite sides. Side a is opposite angle A, side b is opposite angle B, and side c is opposite angle C. This relationship matters because both the Law of Sines and Law of Cosines pair each angle with its opposite side.
An included angle lies between two known sides. An included side lies between two known angles. Correctly identifying what is included separates SAS from SSA and ASA from AAS.
How to Solve an SSS Triangle
SSS provides all three sides. First check the triangle inequality: the sum of every pair must exceed the remaining side. Values 3, 4, and 5 pass because 3 + 4 is greater than 5.
Use the Law of Cosines to find an angle, then find a second angle and subtract both from 180° for the third. A 3-4-5 triangle has angles approximately 36.8699°, 53.1301°, and 90°, so it is a scalene right triangle.
How to Solve an SAS Triangle
SAS supplies two sides and the angle directly between them. Find the opposite side with c² = a² + b² - 2ab cos(C), adjusting labels when another included angle is known.
Once the third side is available, the triangle becomes an SSS problem. Completing it with cosine-law angles avoids the ambiguous inverse-sine branch that can occur when the wrong measurements are treated as SSA.
How to Solve ASA and AAS Triangles
Both ASA and AAS provide two angles, so calculate the third with C = 180° - A - B. The known angles must add to less than 180°. If they do not, no nondegenerate triangle exists.
Next use a/sin(A) = b/sin(B) = c/sin(C). In ASA the known side sits between the two known angles; in AAS it does not. The same sine-law ratios solve both once each side is paired with its opposite angle.
Triangle Area, Heights, and Radii
Area and perimeter
Heron's formula uses the three solved sides: area = √(s(s-a)(s-b)(s-c)), where s is the semiperimeter. Perimeter is a + b + c.
Altitudes
The altitude to side a is 2 × area / a. The same relationship gives the heights to b and c.
Inradius
The incircle touches all three sides. Its radius equals area divided by the semiperimeter.
Circumradius
The circumcircle passes through all vertices. Its radius is abc divided by four times the area.
Triangle Calculator Examples
| Known values | Method | First calculation |
|---|---|---|
| SSS | Law of Cosines | Find an angle |
| SAS | Law of Cosines | Find the third side |
| ASA | Angle sum + Law of Sines | Find the third angle |
| AAS | Angle sum + Law of Sines | Find the third angle |
| AAA | Not uniquely solvable | Scale remains unknown |
| SSA | Ambiguous case | May have 0, 1, or 2 triangles |
Triangle Calculator FAQ
What information is needed to solve a triangle?
Three suitable measurements are required, including at least one side. SSS, SAS, ASA, and AAS each determine one unique triangle.
How do you solve a triangle with three sides?
Check the triangle inequality first. Then use the Law of Cosines to find angles and verify that they total 180°.
How do you solve a triangle with SAS?
Use the included angle in the Law of Cosines to calculate the third side. Complete the remaining angles from the resulting SSS triangle.
What is the difference between ASA and AAS?
ASA's known side lies between the two known angles. AAS uses a side that is not between them, but both use angle sum and the Law of Sines.
Why does SSA sometimes give two triangles?
An inverse-sine result can correspond to an acute angle or its obtuse supplement. Depending on the sides, neither, one, or both possibilities may form a triangle, so SSA needs a separate ambiguous-case solver.
Can three angles determine a triangle?
Three angles determine only the shape and similarity class. Without a side length, the triangle's scale, area, and perimeter remain unknown.
How do you calculate triangle area from three sides?
Find the semiperimeter s = (a+b+c)/2. Then apply Heron's formula √(s(s-a)(s-b)(s-c)).
How do you find the height of a triangle?
After finding area, divide twice the area by the chosen base. For side a, the corresponding altitude is 2 × area / a.
How accurate are the displayed triangle angles?
Calculations retain full floating-point precision internally and round only for display. Small final differences are normal when decimal values are shown.
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