Triangle Calculator

How do you solve a triangle with 3 sides (SSS)?

Use the Law of Cosines: cos(A) = (b² + c² − a²) / (2bc). This gives you angle A. Repeat for B and C, or use the fact that A + B + C = 180°. Once you have all angles, you can calculate area using Heron's formula: Area = √(s(s−a)(s−b)(s−c)), where s is the semi-perimeter (a+b+c)/2.

What is the Law of Sines and when do you use it?

The Law of Sines states that a/sin(A) = b/sin(B) = c/sin(C). Use it when you know an angle-side pair plus one additional value (ASA, AAS, or SSA configurations). It relates each side to the sine of its opposite angle.

How do you calculate the area of a triangle?

There are multiple methods: (1) Heron's formula using three sides: Area = √(s(s−a)(s−b)(s−c)). (2) SAS formula: Area = ½ · a · b · sin(C). (3) Base × height: Area = ½ · base · height. This calculator uses the most appropriate method based on your inputs.

What is the ambiguous case (SSA) in triangle solving?

When given two sides and a non-included angle (SSA), there may be zero, one, or two valid triangles. For example, if a=5, b=8, A=30°, the Law of Sines gives sin(B) = b·sin(A)/a. If sin(B) > 1, no triangle exists. If B could be acute or obtuse, two solutions are possible. This calculator detects and handles all SSA cases.

How do you find the height of a triangle?

The height from side a is hₐ = 2·Area/a. Similarly, h_b = 2·Area/b and h_c = 2·Area/c. First calculate the area using any method (Heron's formula or ½ab·sin(C)), then divide twice the area by each side length.

What is the difference between right, acute, and obtuse triangles?

A right triangle has one angle exactly 90°. An acute triangle has all angles less than 90°. An obtuse triangle has one angle greater than 90°. You can also determine this from sides: if a² + b² = c² (where c is the longest side), it's right; if a² + b² > c², it's acute; if a² + b² < c², it's obtuse.

What are the inradius and circumradius of a triangle?

The inradius (r) is the radius of the largest circle that fits inside the triangle: r = Area / s, where s is the semi-perimeter. The circumradius (R) is the radius of the circle that passes through all three vertices: R = a / (2·sin(A)). Both are useful in geometry and engineering.

Can you solve a triangle with only two angles?

You can find the third angle (since A + B + C = 180°), but you cannot determine the actual side lengths without at least one side. The triangle's shape is determined by its angles, but its size requires a side measurement. This calculator requires at least one side to compute all properties.