SSA Triangle Calculator (Ambiguous Case)
Solve the ambiguous SSA triangle case — enter two sides and a non-included angle to find 0, 1, or 2 possible triangles. Shows all angles, sides, area, and perimeter for each solution.
SSS Triangle Calculator
Solve any triangle from three sides using the Law of Cosines and Heron's formula. Computes all angles, area, perimeter, heights, medians, inradius, and circumradius.
Input — Three Sides
Results
Enter three positive side lengths to solve the triangle.
Reference Table — Common Triangles
| a | b | c | A (°) | B (°) | C (°) | Area | Type |
|---|---|---|---|---|---|---|---|
| 3 | 4 | 5 | 36.87 | 53.13 | 90.00 | 6.00 | Right |
| 5 | 5 | 5 | 60.00 | 60.00 | 60.00 | 10.83 | Acute |
| 5 | 12 | 13 | 22.62 | 67.38 | 90.00 | 30.00 | Right |
| 6 | 8 | 10 | 36.87 | 53.13 | 90.00 | 24.00 | Right |
| 7 | 7 | 7 | 60.00 | 60.00 | 60.00 | 21.22 | Acute |
| 7 | 8 | 9 | 48.19 | 58.41 | 73.40 | 26.83 | Acute |
| 8 | 15 | 17 | 28.07 | 61.93 | 90.00 | 60.00 | Right |
| 10 | 10 | 10 | 60.00 | 60.00 | 60.00 | 43.30 | Acute |
| 10 | 13 | 15 | 41.08 | 58.67 | 80.26 | 64.06 | Acute |
| 13 | 14 | 15 | 53.13 | 59.49 | 67.38 | 84.00 | Acute |
Planning notes, formulas, and examples
About the SSS Triangle Calculator
<p>The <strong>SSS Triangle Calculator</strong> fully solves any triangle when all three side lengths are known. This is the most common real-world scenario — you can measure all three sides of a plot of land, a structural beam triangle, or a geometry homework problem, and the calculator will derive every other property. The only requirement is that the three sides satisfy the <em>triangle inequality</em>: the sum of any two sides must be greater than the third.</p>
<p>The calculation proceeds in two stages. First, the <strong>Law of Cosines</strong> determines each angle: <em>A = arccos((b² + c² − a²) / (2bc))</em>, and similarly for B and C. Second, <strong>Heron's formula</strong> gives the area from the semi-perimeter: <em>Area = √(s(s−a)(s−b)(s−c))</em> where <em>s = (a + b + c) / 2</em>.</p>
<p>Beyond angles and area, this calculator also provides the <strong>three altitudes</strong> (h_a = 2·Area / a), the <strong>three medians</strong>, the <strong>inradius</strong> (radius of the inscribed circle, r = Area / s), and the <strong>circumradius</strong> (radius of the circumscribed circle, R = abc / (4·Area)). Visual bars compare the sides, angles, and heights, and a reference table lists common triangles for quick comparison. Ideal for students, surveyors, engineers, and programmers who need precise triangle computations.</p>
When This Page Helps
When all three sides are known, an SSS solver should do more than confirm the triangle exists. This calculator turns those side lengths into a full geometric profile: all three angles, area, perimeter, altitudes, medians, inradius, and circumradius. That makes it useful not only for geometry assignments, but also for land measurement, fabrication layouts, and any triangle problem where you can measure edges directly but still need the hidden angles and derived dimensions.
How to Use the Inputs
- Enter the three side lengths a, b, and c.
- The calculator checks the triangle inequality automatically.
- View all computed angles, area, perimeter, heights, medians, and radii.
- Use preset buttons to load famous triangles (3-4-5, equilateral, etc.).
- Explore the visual bars to compare sides, angles, and heights.
- Refer to the reference table for common triangle configurations.
Formula used
<p><strong>Law of Cosines:</strong> A = arccos((b² + c² − a²) / (2bc))</p>
<p><strong>Heron's formula:</strong> Area = √(s(s−a)(s−b)(s−c)), where s = (a+b+c)/2</p>
<p><strong>Altitude:</strong> h_a = 2 · Area / a</p>
<p><strong>Inradius:</strong> r = Area / s</p>
<p><strong>Circumradius:</strong> R = abc / (4 · Area)</p>
<p><strong>Median:</strong> m_a = ½√(2b² + 2c² − a²)</p>
Example Calculation
Result: Right scalene triangle with area 6 and angles about 36.87°, 53.13°, and 90°.
<p>Use side lengths <strong>3</strong>, <strong>4</strong>, and <strong>5</strong> in the three input boxes.</p><ul><li>The triangle inequality passes because 3 + 4 > 5</li><li>Semi-perimeter s = (3 + 4 + 5) / 2 = <strong>6</strong></li><li>Heron's formula gives area = √(6 × 3 × 2 × 1) = <strong>6</strong></li><li>Law of Cosines gives angles ≈ <strong>36.87°</strong>, <strong>53.13°</strong>, and <strong>90°</strong></li></ul><p>This classic 3-4-5 input is a quick way to verify that the calculator correctly identifies a right triangle and computes the matching derived measurements.</p>
Tips & Best Practices
- A 3-4-5 triangle is right-angled — use it to quickly verify the calculator.
- An equilateral triangle (a = b = c) always has 60° angles and the largest area-to-perimeter ratio.
- If the triangle inequality fails, increase the shortest side or decrease the longest.
- The circumradius is infinite for degenerate (collinear) triangles — the calculator warns you before that happens.
- For obtuse triangles, the largest angle is always opposite the longest side.
Why SSS Gives A Unique Triangle
If three positive side lengths satisfy the triangle inequality, they determine exactly one triangle up to congruence. That is what makes SSS such a dependable setup. There is no ambiguity about the shape once the three sides are fixed. The only real failure case is invalid input, such as 1, 2, and 5, where the two shorter sides cannot span the longest edge.
This calculator checks that condition first. If the side lengths do form a real triangle, the rest of the geometry follows from those three numbers alone.
From Three Sides To Every Angle And Area
The Law of Cosines converts side lengths into angles. For angle A, use A = arccos((b² + c² − a²) / 2bc), and rotate the letters for B and C. Once the angles are known, the triangle is fully solved. Heron's formula then gives the area directly from the sides: Area = √(s(s − a)(s − b)(s − c)), where s is the semi-perimeter.
For the 3-4-5 example, s = 6 and the area is 6 square units. The same triangle produces angles of about 36.87°, 53.13°, and 90°, confirming it is a right triangle. This is why SSS is so useful in field measurements: you can recover the hidden angles without ever measuring them directly.
What The Extra Outputs Tell You
After solving the basic triangle, the derived measurements become the practical part. Altitudes tell you the perpendicular heights relative to each side. Medians show the distances from vertices to opposite side midpoints. The inradius is useful for inscribed circles and packing problems, while the circumradius is useful when the triangle must fit on a common circle.
Seeing all of those values together helps you compare triangles that may have similar perimeters but very different shapes. That is especially helpful in design layouts, land subdivision sketches, and geometry checks where you need more than just one area value.
Sources & Methodology
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Frequently Asked Questions
SSS means all three sides of a triangle are known. Given three valid side lengths, exactly one triangle is determined (up to congruence).
The sum of any two sides must be strictly greater than the third side: a + b > c, a + c > b, and b + c > a.
Use the Law of Cosines: A = arccos((b² + c² − a²) / (2bc)), and similarly for B and C.
A formula for the area of a triangle given three sides: Area = √(s(s−a)(s−b)(s−c)), where s is the semi-perimeter.
The radius of the largest circle that fits inside the triangle, tangent to all three sides. Computed as r = Area / s.
The radius of the circle passing through all three vertices. Computed as R = abc / (4 · Area).
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