Understanding SSS Triangles
When you know all three sides of a triangle (SSS), you can find all three angles using the Law of Cosines, and the area using Heron's formula. The SSS case always gives a unique triangle (if the sides satisfy the triangle inequality).
Key Formulas
Law of Cosines
Used to find each angle from the three sides.
cos(A) = (b² + c² - a²) / (2bc)
Heron's Formula
Calculate area from three sides using the semi-perimeter.
Area = sqrt(s(s-a)(s-b)(s-c))
Triangle Inequality
A valid triangle requires that any side is less than the sum of the other two.
a + b > c, a + c > b, b + c > a
How to Solve an SSS Triangle
- Verify the triangle inequality: each side must be less than the sum of the other two.
- Use the Law of Cosines to find the largest angle first (opposite the longest side).
- Use the Law of Cosines again for a second angle.
- Find the third angle: A + B + C = 180 degrees.
- Calculate the area using Heron's formula with the semi-perimeter s = (a + b + c) / 2.
Types of Triangles by Sides
- Equilateral: All three sides are equal (a = b = c).
- Isosceles: Two sides are equal.
- Scalene: All three sides are different lengths.