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Sss Triangle Calculator (Side Side Side)

Solve any triangle from three side lengths. Computes area via Heron's formula, all interior angles via the Law of Cosines, perimeter, and semi-perimeter instantly.

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What Is an SSS Triangle Calculator?

An SSS triangle calculator solves every unknown property of a triangle when all three side lengths — side a, side b, and side c — are provided. Because three sides uniquely determine a triangle (provided they satisfy the triangle inequality), this method yields exact values for the area, all three interior angles, the perimeter, and the semi-perimeter without any ambiguity.

Triangle Validity: The Triangle Inequality Theorem

Before applying any formula, the three sides must pass the triangle inequality theorem: the sum of any two sides must exceed the third. All three conditions must hold simultaneously:

  • a + b > c
  • a + c > b
  • b + c > a

If any condition fails — for instance, sides 2, 3, and 9 (since 2 + 3 = 5 < 9) — no triangle exists. The calculator checks this condition automatically before computing any result.

Step 1: Compute the Semi-Perimeter

The semi-perimeter s is the essential first step in Heron's formula and equals half the perimeter:

s = (a + b + c) / 2

For sides a = 5, b = 6, c = 7: s = (5 + 6 + 7) / 2 = 9. The semi-perimeter alone carries no geometric meaning, but it dramatically simplifies the area expression that follows.

Step 2: Calculate the Area Using Heron's Formula

Heron's formula, attributed to Hero of Alexandria (c. 60 AD) and formalized in modern university curricula by Texas A&M University Math 150, computes triangle area directly from three side lengths with no height measurement required:

Area = √[s(s − a)(s − b)(s − c)]

Applying the formula to sides 5, 6, and 7: Area = √(9 × 4 × 3 × 2) = √216 ≈ 14.70 square units. This result is independent of triangle orientation, making Heron's formula the standard choice for programmatic and computational geometry.

Step 3: Determine All Interior Angles Using the Law of Cosines

The Law of Cosines, which generalizes the Pythagorean theorem to any triangle, is rearranged to isolate each angle when all three sides are known. As documented in the UCI Mathematics Math 161 course notes, the three angle formulas are:

  • A = arccos[(b² + c² − a²) / (2bc)]
  • B = arccos[(a² + c² − b²) / (2ac)]
  • C = arccos[(a² + b² − c²) / (2ab)]

For sides a = 5, b = 6, c = 7, the three angles resolve as:

  • A = arccos[(36 + 49 − 25) / 84] = arccos(0.7143) ≈ 44.42°
  • B = arccos[(25 + 49 − 36) / 70] = arccos(0.5429) ≈ 57.12°
  • C = 180° − 44.42° − 57.12° ≈ 78.46°

The three angles always sum to exactly 180°, which serves as a built-in verification of correctness.

Why SSS Always Produces a Unique Triangle

Unlike the SSA configuration — which can produce zero, one, or two valid triangles (the ambiguous case) — SSS always defines exactly one triangle when valid sides are provided. This uniqueness property is formally established in RIT's Academic Success Center handout on Ambiguous Triangles. The absence of ambiguity makes SSS the most reliable and straightforward triangle-solving configuration available.

Practical Applications of SSS Triangle Calculations

SSS triangle calculations appear across a wide range of professional and everyday disciplines:

  • Land surveying: Plot boundaries are determined by physically measuring three distances, then computing angles and area from those values.
  • Structural engineering: Triangular truss members are sized and angled by computing forces from known member lengths.
  • GPS and navigation: Trilateration algorithms — the geometric core of GPS positioning — rely on distance-based triangle solving equivalent to SSS.
  • Architecture: Roof pitch, rafter slope, and ridge-to-wall angles are calculated from measured rafter and ridge lengths before any cutting begins.
  • Computer graphics: Mesh triangles are fully described by their edge lengths, and rendering engines use SSS geometry to compute surface normals and lighting angles.

Reference

Frequently asked questions

What does an SSS triangle calculator compute from three side lengths?
An SSS triangle calculator takes three side lengths (a, b, and c) and computes all remaining triangle properties: the semi-perimeter using s = (a+b+c)/2, the area using Heron's formula, all three interior angles using the Law of Cosines, and the perimeter. For example, sides 5, 6, and 7 produce a semi-perimeter of 9, an area of approximately 14.70 square units, and interior angles of approximately 44.42, 57.12, and 78.46 degrees.
What is Heron's formula and how is it applied in SSS triangle calculations?
Heron's formula calculates triangle area from three side lengths without needing a height measurement. First, compute the semi-perimeter: s = (a+b+c)/2. Then apply Area = sqrt(s(s-a)(s-b)(s-c)). For sides 5, 6, and 7, s = 9 and Area = sqrt(9 x 4 x 3 x 2) = sqrt(216), which is approximately 14.70 square units. Attributed to Hero of Alexandria around 60 AD, the formula works for all triangle types: acute, obtuse, and right.
Can any three numbers be used as side lengths in an SSS triangle calculator?
No. Three values form a valid triangle only when every pair of sides sums to more than the third side — a requirement called the triangle inequality theorem. All three conditions must hold: a+b > c, a+c > b, and b+c > a. For example, sides 1, 2, and 10 fail because 1+2 equals only 3, which is less than 10. A reliable SSS calculator automatically validates this condition and alerts the user when the entered values cannot form a real triangle.
How does the Law of Cosines find all three angles when only side lengths are known?
The Law of Cosines rearranges into three angle formulas: A = arccos((b^2+c^2-a^2)/(2bc)), B = arccos((a^2+c^2-b^2)/(2ac)), and C = arccos((a^2+b^2-c^2)/(2ab)). For sides 5, 6, and 7, angle A is approximately 44.42 degrees, angle B is approximately 57.12 degrees, and angle C is approximately 78.46 degrees, summing to exactly 180 degrees. Texas A&M University's Math 150 materials confirm this rearrangement as the standard method for SSS angle computation.
What real-world fields rely on SSS triangle calculations?
SSS triangle calculations are used in land surveying to determine plot boundaries from three physically measured distances, in structural engineering to compute joint angles and forces in triangular trusses, in GPS trilateration to locate a receiver from its distances to three known satellite positions, and in architecture to calculate roof rafter angles from measured ridge and rafter lengths. Any professional discipline that takes direct distance measurements rather than angle readings benefits from the SSS method's precision and uniqueness.
What is the difference between SSS, SAS, and ASA methods for solving triangles?
SSS (Side-Side-Side) uses all three known sides to compute all angles and the area, always producing exactly one unique triangle when valid inputs are given. SAS (Side-Angle-Side) provides two sides and the included angle, also yielding a unique solution. ASA (Angle-Side-Angle) uses two angles and the included side. SSA (Side-Side-Angle), by contrast, is the ambiguous case that can yield zero, one, or two triangles, as documented by RIT's Academic Success Center. SSS is the most deterministic method available when all physical distances can be measured directly.