1. Enter Known Values (Sides and/or Angles):
Enter up to 3 sides and/or 3 angles. The calculator will attempt to solve the triangle based on valid combinations (e.g., SSS, SAS, ASA, AAS, SSA).
How to Use the Triangle Theorems Calculator
This calculator helps you solve for unknown sides, angles, and other properties of a triangle based on the values you provide. It automatically detects the case (like SSS, SAS, ASA, AAS, or the ambiguous SSA) and performs the necessary calculations.
1. Enter Known Values:
- Input the lengths of any known sides (
Side a,Side b,Side c). - Input the measures of any known angles (
Angle A,Angle B,Angle C) in degrees. Angles must be greater than 0 and less than 180. - You need to provide a valid combination of at least three values for the calculator to solve the triangle. Common valid combinations include:
- SSS: Three sides.
- SAS: Two sides and the angle between them.
- ASA: Two angles and the side between them.
- AAS: Two angles and a non-included side.
- SSA: Two sides and an angle not between them (this is the ambiguous case and may result in 0, 1, or 2 solutions).
2. Solve the Triangle:
- Click the “Solve Triangle” button.
3. Understand the Results:
The calculator will display the results, potentially in one or two “Solution” sections if the SSA case yields multiple triangles. For each valid solution, you’ll see sides, angles, classification, properties, and a visual diagram.
4. Clearing Inputs:
- Click “Clear All” to reset all input fields and results.
Unlocking Triangle Secrets: A Comprehensive Guide to Triangle Theorems & Calculations
The Ubiquitous Triangle: A Foundation of Geometry and Beyond
Triangles. These three-sided polygons are fundamental shapes in geometry, forming the building blocks for more complex structures and appearing everywhere in the natural and man-made world. From the trusses supporting a bridge to the sails of a ship, from the pyramids of ancient civilizations to the pixels on our screens, triangles are a testament to stability and versatility.
Core Triangle Properties: The Ground Rules
Before diving into complex theorems, let’s revisit some basic but essential properties of all triangles: the sum of interior angles is 180 degrees, the Triangle Inequality Theorem must hold (sum of any two sides > third side), and side lengths correspond to their opposite angles (longest side opposite largest angle).
Classifying Triangles: Giving Them Names
Triangles are classified by side lengths (Equilateral, Isosceles, Scalene) and by angle measures (Acute, Right, Obtuse). Our calculator determines these types for you.
Key Theorems for Solving Triangles
“Solving a triangle” means finding all its sides and angles. Key tools include:
1. The Pythagorean Theorem (Right Triangles Only)
a² + b² = c², where c is the hypotenuse.
2. The Law of Sines
a / sin(A) = b / sin(B) = c / sin(C) = 2R (R is circumradius). Used for ASA, AAS, and SSA cases.
3. The Law of Cosines
c² = a² + b² - 2ab * cos(C) (and its permutations). Used for SSS and SAS cases.
Calculating the Area of a Triangle
Methods include 0.5 * base * height, 0.5 * a * b * sin(C) (SAS), and Heron’s Formula (SSS: Area = sqrt(s(s-a)(s-b)(s-c)) where s is the semi-perimeter).
“Geometry is the art of good reasoning from badly drawn figures.” – Henri Poincaré. Our calculator aims to provide both good reasoning (calculations) and a reasonably good figure (SVG diagram)!
Conclusion: Mastering the Triangle
Triangles are more than just simple shapes; they are a gateway to understanding complex geometric relationships and a practical tool in numerous fields. Our Triangle Theorems Calculator is designed to be your companion in this exploration, handling the intricate math and providing clear, comprehensive results.