Triangle Calculator
Triangle Solution Methods
Triangle Formulas and Properties
Basic Properties
- The sum of all angles in a triangle equals 180°
- The sum of any two sides must be greater than the third side
- The longest side is opposite the largest angle
- An equilateral triangle has all sides and angles equal (60° each)
Law of Cosines
c² = a² + b² - 2ab·cos(C)
a² = b² + c² - 2bc·cos(A)
b² = a² + c² - 2ac·cos(B)
Use the Law of Cosines when you know three sides (SSS) or two sides and the included angle (SAS).
Pythagorean Theorem
Special Case: Right Triangles
Where c is the hypotenuse (longest side, opposite the 90° angle) and a, b are the other two sides.
How Law of Cosines Reduces to Pythagorean Theorem
When angle C = 90° (a right triangle), the Law of Cosines simplifies to the Pythagorean theorem:
c² = a² + b² - 2ab·cos(C)
c² = a² + b² - 2ab·cos(90°)
c² = a² + b² - 2ab·(0) ← cos(90°) = 0
c² = a² + b²
This shows that the Pythagorean theorem is actually a special case of the more general Law of Cosines.
For dedicated right triangle calculations with step-by-step solutions, try our Pythagorean Theorem Calculator.
Law of Sines
a/sin(A) = b/sin(B) = c/sin(C)
Use the Law of Sines when you know two angles and any side (ASA, AAS) or two sides and a non-included angle (SSA).
Area Formulas
Base × Height
A = ½ × base × height
Heron's Formula
A = √(s(s-a)(s-b)(s-c))
where s = (a+b+c)/2
Two Sides + Angle
A = ½ × a × b × sin(C)
Equilateral Triangle
A = (√3/4) × side²
Types of Triangles
By Sides:
- Equilateral: All sides equal
- Isosceles: Two sides equal
- Scalene: No sides equal
By Angles:
- Acute: All angles < 90°
- Right: One angle = 90°
- Obtuse: One angle > 90°
Related Calculators
Frequently Asked Questions
What is an SSA triangle calculator?
An SSA triangle calculator solves triangles when you know two sides and a non-included angle (Side-Side-Angle). This is also called the 'ambiguous case' because it may have zero, one, or two valid solutions depending on the given values.
How do I solve a triangle with SSS, SAS, ASA, or AAS?
For SSS (three sides) and SAS (two sides + included angle), use the Law of Cosines. For ASA (two angles + included side) and AAS (two angles + non-included side), use the Law of Sines. Our calculator automatically detects which method to use based on your inputs.
What is the Law of Cosines?
The Law of Cosines states: c² = a² + b² - 2ab·cos(C). It relates the lengths of a triangle's sides to the cosine of one of its angles. Use it when you know three sides (SSS) or two sides and the included angle (SAS).
How do I calculate the area of a triangle?
There are several methods: 1) Base × Height: A = ½ × base × height, 2) Heron's Formula: A = √(s(s-a)(s-b)(s-c)) where s = (a+b+c)/2, 3) Two sides + included angle: A = ½ × a × b × sin(C).
How is the Pythagorean theorem related to the Law of Cosines?
The Pythagorean theorem (c² = a² + b²) is a special case of the Law of Cosines. When angle C equals 90°, the term -2ab·cos(C) becomes zero because cos(90°) = 0, reducing c² = a² + b² - 2ab·cos(C) to simply c² = a² + b². This means the Law of Cosines is a generalization that works for all triangles, while the Pythagorean theorem only applies to right triangles.