Triangle Calculator — Sides Angles Area

Triangle Diagram

Triangle Solver Calculator

Enter at least 3 known values (at least one must be a side). Leave unknown fields blank.

📹 Video Walkthrough — How to Use This Calculator

Mathematical Equations

Law of Sines

This formula links the lengths of sides with the sines of their opposite angles—even in odd-shaped triangles.

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

Law of Cosines

Use this for triangles where you know two sides and the included angle, or all three sides. It’s direct and avoids guesswork.

c² = a² + b² - 2ab·cos(C)
b² = a² + c² - 2ac·cos(B)
a² = b² + c² - 2bc·cos(A)

Heron's Formula for Area

If you have all three sides, this formula gives the area directly—no need for any angles.

s = (a + b + c)/2
Area = √[s(s-a)(s-b)(s-c)]

Alternative Area Formulas

If you know two sides and the included angle instead, use these versions for area—they’re fast and practical on a sketch.

Area = ½ab·sin(C)
Area = ½ac·sin(B)
Area = ½bc·sin(A)

Triangle Theory & Engineering Applications

Triangle solvers aren’t just a math exercise—they’re built into the workflow for just about every field that touches geometry. Whether you’re laying out a bracket, a truss, or a boundary line, triangles show up in force analysis, surveying, linkages, and machining. Practically, triangles let you connect the dots between lengths and angles with minimal fuss. The calculator follows the basic triangle rule: three sides/angles to start, the rest from trig.

Fundamental Triangle Properties

Every triangle has three sides (a, b, c) and three angles (A, B, C). The sum of interior angles is always 180°, and that lets you fill in the blanks once you know enough about a triangle. This is the core idea behind using laws of sines and cosines—it’s practical, not theoretical, and it’s what you use in real design and measurement work.

The law of sines lets you relate lengths and their opposite angles. It’s most direct when you have ASA, AAS, or sometimes SSA info (but with SSA, be ready for the “ambiguous case”—sometimes it means there are two possible solutions physically).

The law of cosines bridges the gap when you don’t have a right triangle. If you’ve got three measured side lengths, or two sides and the angle between, start here. It’s Pythagoras’ theorem with an adjustment for the angle—not just for the textbook, but for real assemblies and field measurements.

Engineering Applications

In mechanical engineering, triangles come up wherever loads and forces are distributed—trusses, linkages, and frame joints all rely on triangles. The underpinning geometry tells you where to drill, where the force lands, and how to size a member. When mounting actuators at angles, expect to use triangle calculations to answer how much stroke you get and what force you actually deliver—these are practical, day-to-day uses.

If you work with linear actuators in an angled setup or a mechanism, triangles appear in the static sketch before you ever cut metal. They define stroke, required force for a job, and clearances. The calculator’s answers feed straight into your mounting dimensions and linkage analysis.

Surveying, GPS, and navigation all lean on triangulation. You get two distances and an angle (or vice versa), and suddenly you know your precise spot on a drawing or map. This isn’t abstract—it shows up every day on-site and in data processing after field work.

Manufacturing and Machining

Triangle calculations set cutter paths, locate holes, and define fixture points. Ever need to figure out an angle from measured points? Or shift between cartesian and angular coordinates for a CNC? This is the triangle math that makes the digital world line up with the physical part.

If you use a CMM or other inspection gear, triangles are there in the math—surveying a part, checking tolerances, and confirming you’re within spec. It helps convert 3D data into meaningful results for pass/fail decisions.

Robotics and Automation

Robotic arm movement is essentially a stack of triangles—each joint, each link, every end effector position. Both forward and inverse kinematics work out by solving sets of triangles. You can’t avoid it in practical layouts or code for path planning.

If you’re working with AGVs or any mobile robot, onboard guidance relies on measuring from known points using triangle math—laser, ultrasound, or vision—all to stay on track or avoid collisions. Again, it’s field-proven geometry, not just classroom stuff.

Worked Example

Problem: Linear Actuator Mounting Analysis

Suppose you’re setting up an actuator to lift a hinged platform. The actuator pivot is 12" from the main pivot, with the platform connection 18" out. You want a max lift angle at the pivot of 45°. Here’s the direct path to line up the mounting and find your required actuator stroke.

Solution Steps:

Step 1: Law of Cosines for required actuator length:

a² = b² + c² - 2bc·cos(C)
a² = 18² + 12² - 2(18)(12)·cos(45°)
a² = 324 + 144 - 432 × 0.7071
a² = 468 - 305.47
a² = 162.53
a = 12.75 inches

Step 2: Law of Sines for the actuator mounting angle:

sin(A)/a = sin(C)/c
sin(A) = a × sin(C)/c
sin(A) = 12.75 × sin(45°)/12
sin(A) = 12.75 × 0.7071/12
sin(A) = 0.7513
A = 48.76°

Step 3: Angle B from sum of angles:

B = 180° - A - C
B = 180° - 48.76° - 45°
B = 86.24°

Step 4: Area swept—plug directly into the ½bc·sin(A) formula:

Area = ½bc·sin(A)
Area = ½ × 18 × 12 × sin(48.76°)
Area = 108 × 0.7513
Area = 81.14 square inches

Frequently Asked Questions

About the Author

Robbie Dickson

Chief Engineer & Founder, FIRGELLI Automations

Robbie Dickson brings over two decades of engineering expertise to FIRGELLI Automations. With a distinguished career at Rolls-Royce, BMW, and Ford, he has deep expertise in mechanical systems, actuator technology, and precision engineering.