Triangle Calculator — All 6 Solvers

Complete Guide to Solving Triangles: Formulas, Methods, and Calculator Logic

Trigonometry is one of the most practical branches of mathematics, used in everything from construction and engineering to GPS navigation and computer graphics. At the heart of trigonometry lies the triangle—a simple three-sided polygon that holds complex mathematical relationships. This guide will walk you through the geometry behind our Triangle Calculator, helping you understand how to solve for unknown sides, angles, area, and perimeter using the standard trigonometric laws.

1. Understanding Triangle Classifications

Before diving into calculations, it helps to identify what kind of triangle you are working with. Triangles are typically classified by their sides and their angles.

Classification by Sides

• Equilateral Triangle: All three sides are equal in length. Consequently, all three internal angles are also equal (60° each).
• Isosceles Triangle: Has at least two sides of equal length. The angles opposite these equal sides are also equal.
• Scalene Triangle: All three sides have different lengths, and all three internal angles are different.

Classification by Angles

• Right Triangle: Contains exactly one 90° angle (a right angle). The side opposite the right angle is called the hypotenuse.
• Acute Triangle: All three angles are less than 90°.
• Obtuse Triangle: Contains one angle greater than 90°.

Regardless of the type, the sum of internal angles in any Euclidean triangle is always 180 degrees (or π radians).

2. Key Trigonometric Formulas Used in This Calculator

Our calculator determines which formula to apply based on the inputs you provide. Here are the primary mathematical laws used in the background.

The Law of Cosines

The Law of Cosines is a generalization of the Pythagorean Theorem applicable to any triangle, not just right-angled ones. It relates the lengths of the sides to the cosine of one of its angles.

Formula: c² = a² + b² - 2ab · cos(C)

This law is essential when you have:

• SSS Case: Three sides known. You can rearrange the formula to solve for angles (e.g., cos(C) = (a² + b² - c²) / 2ab).
• SAS Case: Two sides and the included angle known. You can solve for the third side directly.

The Law of Sines

The Law of Sines states that the ratio of the length of a side to the sine of its opposite angle is constant for all three sides of the triangle.

Formula: a / sin(A) = b / sin(B) = c / sin(C)

This law is used when solving:

• ASA Case: Two angles and the included side are known.
• AAS Case: Two angles and a non-included side are known.
• SSA Case: Two sides and a non-included angle are known (The Ambiguous Case).

Heron's Formula (Area Calculation)

To calculate the area of a triangle when only the three side lengths (a, b, c) are known, we use Heron's Formula. It does not require knowing the height.

First, calculate the semi-perimeter (s): s = (a + b + c) / 2.

Then, the Area = √[s(s-a)(s-b)(s-c)].

3. Detailed Breakdown of Calculator Modes

Geometry problems usually present you with 3 known values. Depending on which values you have, you must select the correct mode in our calculator.

SSS Solver (Side-Side-Side)

Use this mode when you know the lengths of all three sides (a, b, c) but no angles. This is common in construction when checking if a physical structure is square or calculating corner angles for fitting materials.

Calculation Steps: The calculator first checks the "Triangle Inequality Theorem," which states that the sum of any two sides must be greater than the third side (a + b > c). If valid, it applies the Law of Cosines to find the three angles.

SAS Solver (Side-Angle-Side)

Use this mode when you know two sides (e.g., a and b) and the angle between them (Angle C). The angle must be the "included" angle; otherwise, it becomes an SSA problem.

Calculation Steps: The calculator finds side c using the Law of Cosines. Once all three sides are known, it calculates the remaining angles.

ASA Solver (Angle-Side-Angle)

Use this mode when you know two angles and the side connecting them (the included side). For example, if you are surveying land and measure two corners and the distance between them.

Calculation Steps: Since the sum of angles is 180°, the third angle is easily found (180° - A - B). Then, the Law of Sines is used to find the remaining two sides.

AAS Solver (Angle-Angle-Side)

Similar to ASA, but the known side is not between the two known angles. For example, knowing Angle A, Angle B, and Side a.

Calculation Steps: The third angle is calculated first (Angle C = 180° - A - B). Then the Law of Sines is applied to find sides b and c.

SSA Solver (The Ambiguous Case)

This is the most complex scenario in triangle geometry. You know two sides and an angle that is NOT between them (e.g., Side a, Side b, and Angle A). This is called "ambiguous" because three outcomes are possible:

1. No Solution: The known side is too short to reach the other side, meaning no triangle can be formed.
2. One Solution: A unique triangle is formed (often a right triangle).
3. Two Solutions: The known side can swing in two directions, creating two valid triangles with different shapes (one acute, one obtuse).

Our calculator automatically detects these scenarios. If two solutions exist, a dropdown menu will appear allowing you to toggle between "Solution 1" and "Solution 2".

4. Calculating Area, Perimeter, and Heights

Beyond finding missing sides and angles, our tool provides three other critical properties:

• Perimeter: The total distance around the triangle (P = a + b + c). Useful for fencing and framing.
• Area: The amount of space inside the triangle. If you used the SSS solver, Heron's formula is used. For SAS, the formula Area = 0.5 * a * b * sin(C) is used.
• Heights (Altitudes): The perpendicular distance from a vertex to the opposite side. The calculator provides three heights (ha, hb, hc), corresponding to base a, b, and c respectively. Formula: Height_a = (2 * Area) / a.

5. Common Geometry Errors & Troubleshooting

If you are getting an error message or unexpected result, check for these common issues:

• Triangle Inequality Violation: In an SSS problem, if you enter sides 1, 2, and 10, the calculator will return an error. A triangle cannot be closed if one side is longer than the other two combined.
• Sum of Angles > 180: In ASA or AAS, if the two angles you enter add up to 180° or more, a triangle cannot be formed (since the third angle would be 0 or negative).
• Unit Confusion: Ensure you are using the correct setting for angles. Degrees (deg) are typically used in K-12 education and construction, while Radians (rad) are used in advanced calculus and physics. 180 degrees = π radians (approx 3.14159).

Frequently Asked Questions (FAQ)

Real-World Applications

Architecture & Construction: Calculating roof pitches, truss angles, and floor layouts relies heavily on SSS and SAS calculations.

Surveying: Measuring the distance between two inaccessible points (like across a river) often uses the Law of Sines (ASA/AAS methods).

Machining & CNC: Determining tool paths and cutting angles requires precise trigonometric solutions to ensure parts fit together correctly.