Heron's Formula:
Area = √[s(s - a)(s - b)(s - c)] where s = (a + b + c)/2
Law of Cosines:
c² = a² + b² - 2ab·cos(γ)
Law of Sines:
a/sin(α) = b/sin(β) = c/sin(γ) = 2R
Understanding Triangle Geometry
A triangle is a fundamental three-sided polygon in Euclidean geometry. The sum of all interior angles always equals exactly 180 degrees (π radians). Depending on which three parameters are known, different geometric laws and trigonometric identities allow you to solve for all remaining sides, angles, and properties.
Key Equations
- Sum of Angles:
α + β + γ = 180°
- Inradius (r):
r = Area / s
- Circumradius (R):
R = (a × b × c) / (4 × Area)
Frequently Asked Questions
What is Heron's Formula?
Heron's formula allows calculating triangle area directly from three side lengths without requiring any angle or height measurement: Area = √[s(s - a)(s - b)(s - c)], where s is the semi-perimeter (a + b + c) / 2.
What is the Triangle Inequality Theorem?
A triangle can only exist if the sum of any two sides is strictly greater than the third side: a + b > c, a + c > b, and b + c > a.