SSS Triangle Calculator
Solve Side-Side-Side triangles. Enter three sides to find all angles and triangle properties.
Enter SSS Values
SSS Triangle:
All three sides are known. The calculator will find all three angles using the Law of Cosines and verify the triangle is valid.
Solution Method:
1. Verify triangle inequality
2. Use Law of Cosines to find all angles
3. Calculate area using Heron's formula
Frequently Asked Questions
What is a SSS triangle?
SSS stands for Side-Side-Side. It's a triangle where all three side lengths are known. This uniquely determines the triangle - there's only one possible triangle (up to reflection) with any given set of three side lengths.
How do you solve a SSS triangle?
First, verify the triangle inequality (sum of any two sides must be greater than the third). Then use the Law of Cosines three times to find each angle: A = arccos[(b² + c² - a²)/(2bc)], and similarly for angles B and C.
What is the triangle inequality?
The triangle inequality states that the sum of any two sides of a triangle must be strictly greater than the third side. This must be true for all three combinations: a + b > c, a + c > b, and b + c > a. If any of these fail, no triangle can exist.
Why can't I use the Law of Sines for SSS?
The Law of Sines requires knowing at least one angle to start with. In SSS, you don't know any angles initially - that's what you're trying to find! The Law of Cosines is specifically designed to find angles when you know all three sides.
What is Heron's formula?
Heron's formula calculates the area of a triangle from its three side lengths: Area = √[s(s-a)(s-b)(s-c)], where s is the semiperimeter (s = (a+b+c)/2). This is particularly useful for SSS triangles where you don't initially know any angles.
How can I tell if my triangle is acute, right, or obtuse?
After calculating all angles: if all angles are less than 90°, it's acute; if one angle equals 90°, it's right; if one angle is greater than 90°, it's obtuse. You can also check before calculating: if c² = a² + b², it's right; if c² > a² + b², it's obtuse; if c² < a² + b², it's acute (where c is the longest side).
Can SSS ever have no solution?
Yes, if the sides don't satisfy the triangle inequality. For example, sides of 2, 3, and 10 cannot form a triangle because 2 + 3 = 5 is not greater than 10. The calculator will detect this and show an error message.
Why should the angles sum to exactly 180°?
In Euclidean (flat) geometry, the sum of angles in any triangle is always exactly 180°. Our calculator verifies this as a check on the calculations. If the sum differs significantly from 180°, there may be a calculation error or the input values may be invalid.