How to Use the Triangle Calculator
Our comprehensive triangle solver makes it easy to find all the properties of a triangle, regardless of what information you start with. Follow these simple steps:
- Select Input Type: From the dropdown menu, choose the combination of data you know. The options correspond to standard geometric cases:
- SSS: You know the lengths of all three sides (a, b, c).
- SAS: You know two sides and the angle between them (e.g., side a, angle C, side b).
- ASA: You know two angles and the side between them (e.g., angle A, side c, angle B).
- AAS: You know two angles and a non-included side (e.g., angle A, angle B, side a).
- Enter Values: The calculator will automatically enable the required input fields. Enter your known lengths and/or angles (in degrees).
- Calculate: Click the "Calculate" button. The tool will solve for all missing properties and display them in the results section.
The Mathematics of Solving a Triangle
This calculator uses fundamental trigonometric laws to solve triangles. Depending on the input, it applies different formulas to find the missing values.
The Law of Cosines
The Law of Cosines relates the lengths of the sides of a triangle to the cosine of one of its angles. It's essential for solving SSS and SAS cases.
c² = a² + b² - 2ab * cos(C)
It can be rearranged to find an angle when all three sides are known:
C = arccos((a² + b² - c²) / 2ab)
The Law of Sines
The Law of Sines is used for ASA and AAS cases. It establishes a relationship between the sides of a triangle and the sines of their opposite angles.
a / sin(A) = b / sin(B) = c / sin(C)
Once one side and its opposite angle are known, this law can be used to find the remaining sides or angles.
Area Calculation
The calculator uses two primary methods to find the area:
- Heron's Formula (for SSS): When all three sides are known, the area can be found using the semi-perimeter (s = (a+b+c)/2). Area =
√[s(s-a)(s-b)(s-c)]. - Trigonometric Formula (for SAS): When two sides and the included angle are known. Area =
0.5 * a * b * sin(C).
Triangle Classification
Our calculator also classifies the triangle for you based on its properties:
- By Side Length:
- Equilateral: All three sides are equal.
- Isosceles: Two sides are equal.
- Scalene: No sides are equal.
- By Internal Angles:
- Right Triangle: One angle is exactly 90°. You might also find our Pythagorean Theorem Calculator useful for these.
- Obtuse Triangle: One angle is greater than 90°.
- Acute Triangle: All three angles are less than 90°.
For more tools like this, explore our main Math Calculators page.