Triangle Solver All Cases Interactive Calculator

If you know three measurements of a triangle—sides, angles, or a mix—that's enough to find everything else, but you have to match your method to the case at hand (SSS, SAS, ASA, AAS, or SSA). Get this wrong, and your answers won't add up, especially in the notorious SSA case where you can get two triangles instead of one. The calculator below lets you input any three values you know and handles the number crunching to fill in the rest—lengths, angles, area, perimeter. This is the kind of geometry you run into in real engineering: frame analysis, surveying, path-finding, or even something like a simple 2-joint robot arm. Down the page, you'll find the key formulas, a worked example pulled from actual shop drawings, detailed explanations of every method, and an FAQ that points out the common sticking points—like how ambiguous cases pop up, and what level of precision is actually practical for field work.

What is triangle solving?

Triangle solving is just working out all the unknown side lengths and angles if you know three of the six values. If you tell the calculator any three of those—doesn’t matter if it’s sides or angles—it’ll run the right trigonometry and work out the rest.

Simple Explanation

A triangle only has six main values: three sides and three angles. If you lock in any three of those (with at least one side), the rest are fixed, and the rest can be calculated. The calculator checks what you’ve got and picks from the Law of Sines, Law of Cosines, or the angle sum, depending on your inputs.

📐 Browse all 1000+ Interactive Calculators

Triangle Diagram

How to Use This Calculator

1. Select your triangle case from the dropdown — SSS, SAS, ASA, AAS, SSA, or Right Triangle.
2. Enter the known side lengths in the units of your choice (meters, inches, feet — consistent units only).
3. Enter any known angle values in degrees.
4. Click Calculate to see your result.

Triangle Solver Calculator

Triangle Solving Equations

If you know two sides and the angle in between, use this version of the Law of Cosines:

If you've got two angles and a side, you'll need the Law of Sines:

Once you know two angles, the third is always 180° minus the other two:

To get the area from just side lengths, Heron's formula works:

For right triangles, the Pythagorean theorem gives you the missing side:

Simple Example

Case: SSS — Three sides known
Side a = 3, Side b = 4, Side c = 5
Angle A = arccos((16 + 25 − 9) / 40) = 36.87°
Angle B = arccos((9 + 25 − 16) / 30) = 53.13°
Angle C = 180 − 36.87 − 53.13 = 90.00°
Area = 6.00 square units, Perimeter = 12.00 units

Theory & Engineering Applications

Fundamental Triangle Solving Methods

Solving a triangle is about working out all six details—three sides and three angles—given any three you already know (as long as at least one’s a side). You’ll get nowhere trying to use the wrong formula for your situation: the Law of Cosines covers “all sides known” (SSS) or “two sides and included angle” (SAS). The Law of Sines comes into play if you have a side with two angles (AAS or ASA), or two sides and an angle not in between them (SSA). The sum of angles saves time—if you know two angles, just subtract from 180° to get the third.

The main pitfall is the ambiguous SSA case. If you know two sides and an angle opposite one of them, the Law of Sines sometimes spits out two possible triangles, sometimes one, sometimes none. This happens mostly when the known angle is acute and its opposite side is shorter than the adjacent side. You’ll need extra info (like drawings or practical context) to figure out which triangle actually fits your physical scenario—classic issue in surveying and navigation work.

The Law of Cosines: Derivation and Application

The Law of Cosines is a direct extension of Pythagoras, but for triangles that aren’t right-angled. Let’s say you’ve got a triangle with sides a, b, c, and you know angle C opposite side c. The relationship is c² = a² + b² − 2ab·cos(C). If that angle C is 90°, cos(90°) is zero, and you’re right back to Pythagoras. Watch the minus sign: for acute angles the third side is a bit shorter than naïvely adding a² + b², and for obtuse angles it’s longer.

Practical example: you’re checking if a truss member will fit. The bottom chord is 8.5 m, the slope is 6.2 m, and the angle between them is 127°. Using c² = a² + b² − 2ab·cos(C): c² = 72.25 + 38.44 − 105.4·(−0.6018). That last term turns positive, and you get c = 13.20 meters. That’s the number the shop needs for cutting, and you use it to look up allowable stress and member deflection after the fact.

The Law of Sines and Angular Solutions

The Law of Sines simply says the ratio of a triangle’s side to the sine of its opposite angle is constant: a/sin(A) = b/sin(B) = c/sin(C). This isn’t just elegant—it’s exactly how you get unknown sides if you have any two angles and a side (ASA, AAS) or, if handled carefully, unknown angles from two sides and a non-included angle (SSA). It comes down to relating area: (1/2)ab·sin(C), (1/2)bc·sin(A), (1/2)ac·sin(B)—set those equal, follow the algebra, and you get the Law of Sines.

Picture a survey job: baseline is 47.3 m, measured angles are 53.8° and 71.4°. The last angle is 54.8°. You want the distance across a river: d/sin(71.4°) = 47.3/sin(54.8°), so d = 54.87 m. To find the width directly across, use d times sin(53.8°) and you get 44.22 m. All done without setting foot in the river.

Worked Example: Complete Triangle Solution for Structural Analysis

Say you’re fabricating a steel brace—two segments known, angle between known, and you need everything else for the cut sheet (SAS case): a = 3.75 m, b = 2.90 m, angle C = 68.5°.

Step 1: Side c using Law of Cosines:
c² = (3.75)² + (2.90)² − 2(3.75)(2.90)·cos(68.5°)
c² = 14.0625 + 8.41 − 7.971
c² = 14.5015
c = 3.808 m

Step 2: Angle A with Law of Sines:
sin(A) = 3.75·sin(68.5°)/3.808 = 0.9164
A = arcsin(0.9164) = 66.43°

Step 3: Angle B by subtraction:
B = 180° − 66.43° − 68.5° = 45.07°

Step 4: Perimeter = 3.75 + 2.90 + 3.808 = 10.458 m

Step 5: Area via Heron’s formula:
s = 10.458/2 = 5.229 m
Area = √[5.229·1.479·2.329·1.421] = √[25.426] = 5.042 m²

Now you’ve got cut sizes, weld angles, the total material needed, and the coated surface. Nothing magical, just applying the right routines so the fabricators get exactly what they need and you keep errors down to a minimum.

Engineering Applications Across Industries

In aerospace, solving basic triangles lets you break down force paths through an airframe, especially in wing structures where each truss member turns into an SSS problem with unique angles that feed directly into stress and deflection checks. Say you’ve got a triangle of members—two are fixed lengths, and the diagonal is measured—now you solve angles and distribute loads correctly at your connections.

Surveying is straight-up triangle solving in the wild. A control network is just a series of triangles, each solved for distance based on at least one good baseline and a string of angle measurements (often with a total station). You use the Law of Sines repeatedly until you’ve mapped out all your points—even when you’ll never reach some of them in person.

Short-range navigation (coastal, not intercontinental) uses basic plane triangles, measuring bearings and known distances to determine your actual position. You punch your measurements into the triangle solver and find where you are, without waiting for a GPS fix.

For robotics or CNC, triangles map directly to joint angles. A two-link robot arm with known segment lengths and a required reach forms a triangle: you use SSS to get the angles, then program the joints accordingly. If the robot’s segments are 0.45 m and 0.38 m and it needs to reach 0.60 m, the Law of Cosines tells you exactly how much the “elbow” has to bend.

Geodesy and large-scale mapping usually mean spherical triangles (not plane), but the underlying logic is the same: with enough knowns, solve for everything else. Whole national coordinate systems rest on this math, and engineers keep referencing these calculations long after initial surveys are done.

If you’re spending time with engineering calculators, knowing how and when to apply these triangle routines is one of those core skills—doesn’t matter if you’re designing a bracket, laying out a field, or programming a robot.

Practical Applications

Frequently Asked Questions

📹 Video Walkthrough — Triangle Solver All Cases Interactive Calculator

📹 Video Walkthrough — Triangle Solver All Cases Interactive Calculator