Triangle Calculator

Free Online Triangle Calculator (SSS)

This triangle calculator solves for all three angles, the area, and the perimeter when you know the lengths of all three sides — the Side-Side-Side, or SSS, case. Just type in sides a, b, and c, and the calculator instantly returns every other property of the triangle, complete with a scaled diagram and a full step-by-step solution showing exactly how each value was derived.

Whether you're a student verifying a geometry homework answer, a teacher preparing a worked example, a carpenter or surveyor checking a real-world measurement, or an engineer confirming a structural angle, this tool covers the complete SSS triangle-solving workflow in one place — no manual algebra, no separate calculator for angles and another for area.

Triangle Formulas Used

Solving a triangle from three known side lengths relies on two classic, well-established formulas from trigonometry and geometry. Both are applied automatically inside this calculator, and each step is shown separately so you can follow the logic.

• Law of Cosines: c² = a² + b² − 2ab·cos(C) — rearranged to solve for each angle when all three sides are known
• Heron's Formula: Area = √(s(s−a)(s−b)(s−c)), where s is the semi-perimeter — used to find the area without needing any angle first
• Semi-perimeter: s = (a + b + c) / 2 — half the total perimeter, and a required input for Heron's Formula
• Angle Sum Property: A + B + C = 180° — used to find the third angle quickly once two angles are known
• Triangle Inequality Theorem: the sum of any two sides must exceed the third side, or no valid triangle exists

How to Use This Triangle Calculator

Using the calculator takes only two steps. First, enter the lengths of all three sides — labeled a, b, and c — into the input fields. Second, read the results: the calculator instantly displays Angle A (opposite side a), Angle B (opposite side b), Angle C (opposite side c), the Area, and the Perimeter, alongside a to-scale diagram with each vertex and side clearly labeled.

Behind the scenes, the calculator first checks the triangle inequality to confirm your three side lengths can actually form a valid triangle. If they can't — for example, if one side is longer than the sum of the other two — it tells you immediately instead of returning a meaningless result. If the sides are valid, it works through the semi-perimeter, Heron's Formula for area, the Law of Cosines for two of the angles, and the angle sum property for the third — every one of these steps is broken out in the step-by-step solution below the results, so you can see exactly where each number came from.

Worked Example

Suppose a triangular plot of land has sides measuring 5 meters, 6 meters, and 7 meters. Entering a = 5, b = 6, c = 7 gives a semi-perimeter of s = (5 + 6 + 7) / 2 = 9. Applying Heron's Formula, Area = √(9 × 4 × 3 × 2) = √216 ≈ 14.70 square meters.

For the angles, the Law of Cosines gives Angle A ≈ 44.42°, Angle B ≈ 57.12°, and by subtracting from 180°, Angle C ≈ 78.46°. The perimeter is simply 5 + 6 + 7 = 18 meters. This same process works for any three positive side lengths that satisfy the triangle inequality, whether you're solving a textbook SSS problem or measuring an irregular plot in the field.

Types of Triangles This Calculator Can Identify

Once the three angles are known, the shape of the triangle becomes clear. If all three angles are less than 90°, the triangle is acute. If one angle is exactly 90°, it's a right triangle, and if one angle is greater than 90°, it's obtuse. Comparing the side lengths also tells you whether the triangle is scalene (all sides different), isosceles (two sides equal), or equilateral (all three sides equal, with every angle at 60°). The calculator's diagram is drawn to scale, so you can often see the general shape at a glance before even reading the numeric angle values.

Understanding the Diagram

The diagram above is drawn to scale using the exact side lengths you entered, so the proportions of the triangle you see are the actual proportions of your triangle — a long, thin triangle will look long and thin, and a triangle close to equilateral will look close to equilateral. Each vertex is labeled A, B, and C, and each side is labeled with its length directly on the edge it belongs to: side a runs between vertices B and C, side b runs between vertices C and A, and side c runs between vertices A and B.

Remember the naming convention used throughout this calculator: every side is named after the vertex it sits opposite. So Angle A is always the angle at the vertex opposite side a, Angle B is opposite side b, and Angle C is opposite side c. Keeping this in mind makes it much easier to match the diagram to the numeric results in the panel above it.

Tips for Accurate Results

Since this calculator solves triangles purely from side lengths, the accuracy of your results depends entirely on the accuracy of the three measurements you enter. When measuring a real object or plot of land, use a consistent unit for all three sides — mixing meters and feet will produce a nonsensical result even if the triangle inequality check still passes.

It's also worth double-checking your inputs if the calculator reports an invalid triangle. A common cause is a simple typo, such as entering 50 instead of 5, which can easily make one side longer than the sum of the other two. Small rounding differences in the angles are normal and expected, since the calculator works with decimal approximations of π and inverse trigonometric functions, just like a scientific calculator would.

Where Triangle Calculations Are Used

Solving triangles from side lengths is one of the most practical skills in geometry, and it shows up constantly outside the classroom:

• Students and teachers use SSS triangle solving for geometry, trigonometry, and mensuration coursework and exams.
• Civil engineers and surveyors use it to calculate land boundaries, plot areas, and structural angles from field-measured distances.
• Architects and carpenters rely on it to verify roof pitches, trusses, and triangular framing components before cutting material.
• Navigation and mapping applications use triangle-solving principles (triangulation) to determine distances and positions from known reference points.
• DIY and craft projects — from building a triangular garden bed to cutting fabric or wood into a precise triangular shape — benefit from knowing the exact angles and area in advance.