Sss Triangle Calculator - Three Sides to Angles

An sss triangle calculator solves a triangle when you know the lengths of all three sides and nothing else. The "SSS" stands for Side-Side-Side, the case where three side lengths uniquely determine the shape of a Euclidean triangle. It uses the Law of Cosines for the three interior angles, Heron's formula for the area, and the largest angle to label the triangle as acute, right, or obtuse. It fits surveying, construction layout, woodworking, and any geometry problem where the only data you have is three measured edges.

• • Surveying a triangular plot: Confirm the boundary angles and the area after measuring all three edges.
• • Building a frame or truss: Pick the cut angles for rafters, braces, or gussets once the three side lengths are set.
• • Checking a triangle homework problem: Verify the angles and area you computed by hand for a 3-4-5, 5-12-13, or other textbook triangle.
• • Reverse-engineering a real triangle: Measure a physical triangle with calipers or a tape, then recover the angles and area for a drawing, model, or report.

Three side lengths fix the size and the shape of a triangle at the same time, which is the SSS property. Once the three sides are in, the angles, area, and perimeter are all determined.

Enter the three sides in any positive length unit. The sss triangle calculator shows the three interior angles in degrees, the area in the square of the same unit, the perimeter in the original unit, and a label for acute, right, or obtuse. The form rejects inputs that violate the triangle inequality.

For the same three sides a, b, c plus a known included angle, the ABC triangle calculator covers that related case in the same math-conversion family.

The sss triangle calculator takes the three side lengths, validates that they form a real triangle, and runs the Law of Cosines for each angle and Heron's formula for the area.

• a, b, c: The three side lengths you enter. The label a, b, c is arbitrary, but the convention is that a is opposite angle A, b is opposite angle B, and c is opposite angle C.
• s: The semiperimeter, equal to half of the perimeter. Heron's formula uses s along with s−a, s−b, and s−c to build the area under the square root.
• A, B, C: The three interior angles, returned in degrees. They are computed from the Law of Cosines using the side opposite each angle and the other two sides.

The Law of Cosines is the only trigonometric identity required. cos of the angle equals the sum of the squares of the two adjacent sides minus the square of the opposite side, divided by twice the product of the two adjacent sides. Apply it three times, once per angle, and you have the full angle set.

Heron's formula is a closed-form area expression that uses only the three side lengths. It builds the semiperimeter s = (a + b + c) / 2 first, then takes the square root of s times (s−a) times (s−b) times (s−c).

According to Wikipedia, Law of cosines, the Law of Cosines states that for any triangle with sides a, b, c and opposite angles A, B, C, the relationship c^2 = a^2 + b^2 - 2ab·cos(C) holds, which rearranges to cos(C) = (a^2 + b^2 - c^2) / (2ab)

If you already have the angles measured and only one side, the AAA triangle calculator does the inverse problem with the Law of Sines.

These four ideas cover the formulas and the geometry that turn three side lengths into a complete description of the triangle.

Putting the four concepts together turns three side lengths into a complete triangle. The triangle inequality gates the calculation, the Law of Cosines produces the angles, and Heron's formula produces the area.

According to Wikipedia, Heron's formula, the area of a triangle with sides a, b, c and semiperimeter s = (a + b + c) / 2 is given by area = sqrt(s(s − a)(s − b)(s − c)).

When you want a deeper look at the Heron's-formula area step in isolation, the triangle area calculator focuses on that single result from three sides.

Follow these steps in order. The result panel updates as you type, so the form also works as a scratch pad while you try different combinations.

1. 1 Enter side a: Type the first side length in any positive length unit. The default is 7, and any value greater than 0 is accepted.
2. 2 Enter side b: Type the second side length. Use the same unit as side a so the angles, area, and perimeter all line up.
3. 3 Enter side c: Type the third side length. The form rejects the input if the three sides do not satisfy the triangle inequality.
4. 4 Read the triangle type: Check the type label. Acute, right, and obtuse tell you whether the largest angle is below, equal to, or above 90°.
5. 5 Read the three angles: Use the three angle values for cut lists, bearings, or layout. They sum to 180° within rounding error.
6. 6 Use the area and perimeter: Copy the area for material or finish estimates and the perimeter for trim, fencing, or framing totals.

A landscape designer measures a triangular garden bed and gets sides of 8 m, 11 m, and 13 m. Entering a = 8, b = 11, c = 13 gives an obtuse triangle with area 43.81 sq m and perimeter 32 m. The largest angle is just over 90°.

If your result lands on a 90 degree angle, the right triangle calculator covers the Pythagorean shortcuts and right-triangle trig ratios for that special case.

Solving SSS triangles by hand is straightforward once you know the Law of Cosines, but the sss triangle calculator removes a few classes of error that are easy to make on paper.

• • Catches impossible side combinations: If the three sides violate the triangle inequality, the calculator flags the error before you try to take a square root of a negative number in Heron's formula.
• • Solves the full SSS case in one pass: You get all three angles, the area, the perimeter, and the triangle type from a single form submission, instead of running the Law of Cosines three times by hand.
• • Labels the triangle as acute, right, or obtuse: A quick glance at the type label tells you whether the longest side sits across from a sharp, exact, or wide angle, which matters for cut lists and load planning.
• • Works for any valid SSS triangle: The same formulas cover equilateral, isosceles, scalene, acute, right, and obtuse triangles without a separate code path.

These benefits matter most when the inputs come from a real measurement rather than a textbook. Surveyors, drafters, and model builders all get side data from a tape, a laser, or a CAD file.

Three things change the answer the calculator returns, and the limitations below cover the assumptions behind the Law of Cosines and Heron's formula.

• • SSS information only works for planar Euclidean triangles. It does not apply to spherical triangles (such as triangles on a globe drawn from longitude and latitude) where a different set of identities is required.
• • Floating-point rounding means the three angles may not sum to exactly 180° on the display. Treat each angle as a rounded value and assume the true sum is 180° within the last decimal place.

These factors are why the calculator asks for all three sides. Three side lengths are the minimum data that determines a unique Euclidean triangle, and the SSS case is the most direct path from measurement to a complete description.

According to Wolfram MathWorld, Heron's formula, Heron's formula yields the area of a triangle purely from its three side lengths and requires the triangle inequality a + b > c, b + c > a, a + c > b to produce a positive real area

If you ever need to flip between cases (SSS, SAS, ASA, AAS, or AAA) without re-entering values, the general triangle calculator handles all five behind one form.