SSS Triangle Calculator
Solve SSS triangles by entering three side lengths. Free calculator finds angles, area, and perimeter using the law of cosines and Heron formula instantly.
About This Calculator
The SSS Triangle Calculator solves any triangle where all three side lengths are known (Side-Side-Side). Enter the lengths of sides A, B, and C, and the calculator instantly computes the three interior angles using the law of cosines, the area using Heron's formula, and validates whether the three sides satisfy the triangle inequality theorem. This tool is essential for geometry students, engineers, architects, construction professionals, and anyone working with triangular measurements.
Methodology
When three side lengths a, b, and c are known, the SSS triangle is solved using two key formulas:
- Law of Cosines: For each angle, apply the inverse cosine of the ratio of side squares. Angle A = arccos((b^2 + c^2 - a^2)/(2bc)), Angle B = arccos((a^2 + c^2 - b^2)/(2ac)), and Angle C = 180 deg - Angle A - Angle B.
- Heron's Formula for Area: Compute the semi-perimeter s = (a + b + c)/2, then Area = sqrt(s(s - a)(s - b)(s - c)).
- Triangle Inequality Check: A valid triangle requires a + b > c, a + c > b, and b + c > a. If any condition fails, the sides cannot form a triangle.
Key Features
- Compute all three interior angles (alpha, beta, gamma) in degrees with two decimal precision
- Calculate triangle area using Heron's formula
- Validate triangle inequality and identify invalid side combinations
- Shareable URL links -- bookmark or share your triangle results
Regional Notes
Triangle geometry is universal and the SSS calculator uses standard mathematical formulas that apply worldwide. The law of cosines and Heron's formula are used in mathematics curricula across India (CBSE/ICSE), the United States (Common Core), and the United Kingdom (GCSE/A-Level). Whether you use metric units (meters, centimeters) or imperial units (feet, inches), the calculator works with any consistent unit system since all results are dimensionless or in square units matching your input.
Frequently Asked Questions
What is an SSS triangle?
An SSS triangle is a triangle where all three side lengths are known. SSS stands for Side-Side-Side. When you know all three sides, you can compute the three interior angles using the law of cosines and the area using Heron's formula.
How do you find the angles of an SSS triangle?
To find the angles of an SSS triangle, apply the law of cosines: angle A = arccos((b^2 + c^2 - a^2)/(2bc)), angle B = arccos((a^2 + c^2 - b^2)/(2ac)), and angle C = 180 deg - angle A - angle B. The law of cosines relates each angle to the three side lengths uniquely.
What is the area formula for an SSS triangle?
The area of an SSS triangle is calculated using Heron's formula: Area = sqrt(s(s - a)(s - b)(s - c)), where s is the semi-perimeter equal to (a + b + c)/2. This formula works for any triangle when all three side lengths are known.
Can any three side lengths form a triangle?
No, three side lengths form a valid triangle only if they satisfy the triangle inequality theorem: the sum of any two sides must be greater than the third side. Specifically, a + b > c, a + c > b, and b + c > a must all hold true for a valid triangle.
Are SSS triangles always congruent?
Yes, if two triangles have all three sides equal in length, they are congruent by the SSS congruence criterion. This means one triangle can be transformed into the other through rotation, reflection, or translation -- the shape and size are identical.
What is the law of cosines and how is it used for SSS triangles?
The law of cosines states that for any triangle with sides a, b, c and opposing angles A, B, C: a^2 = b^2 + c^2 - 2bc·cos(A). For SSS triangle solving, it is rearranged to cos(A) = (b^2 + c^2 - a^2)/(2bc). This gives each angle uniquely since the cosine function is one-to-one between 0 deg and 180 deg.
Is this SSS triangle calculator free?
Yes, this SSS triangle calculator is completely free to use online. There are no subscriptions, registration requirements, or usage limits. You can solve unlimited SSS triangles for geometry homework, engineering problems, construction projects, or any other application.
What units does the SSS triangle calculator use?
The SSS triangle calculator works with any consistent unit of measurement. Enter side A, side B, and side C all in the same unit (meters, centimeters, feet, inches, etc.). The area result will be in square units matching your input, and the angles are always in degrees.