Oblique Triangle Calculator
Solve a triangle from SSS, SAS, ASA, or AAS data and export the current solution as a validated Excel workbook.
Excel export is ready for the demonstration values.
Known values
Choose the information you know. Side labels follow the standard convention: side a is opposite angle α, b is opposite β, and c is opposite γ.
Solved triangle
SSS: angles use the law of cosines; area uses Heron's formula.
Solution details
| Quantity | Value | Interpretation |
|---|---|---|
| Side a | 5 cm | Opposite angle α |
| Side b | 7 cm | Opposite angle β |
| Side c | 8 cm | Opposite angle γ |
| Angle α | 38.2132° | Interior angle opposite side a |
| Angle β | 60° | Interior angle opposite side b |
| Angle γ | 81.7868° | Interior angle opposite side c |
| Area | 17.320508 cm² | Region enclosed by the three sides |
| Perimeter | 20 cm | Total boundary length |
All displayed values are rounded for readability. The Excel file stores the unrounded finite model values and applies spreadsheet number formats.
How to use the oblique triangle calculator
What this calculator does. It solves one non-degenerate triangle when you provide a sufficient SSS, SAS, ASA, or AAS data set. The solver returns all three side lengths, all three interior angles, area, perimeter, and semiperimeter. It is designed for geometry exercises, layout checks, surveying-style calculations, and dimensional planning. It does not resolve the ambiguous SSA case, because two sides and a non-included angle can describe zero, one, or two different triangles. The underlying methods are the law of sines, the law of cosines, and Heron's formula; OpenStax gives a detailed treatment of both the law of sines for non-right triangles and the related data patterns.
When to use it. Use the calculator when checking whether three measured sides can close into a triangle, finding a missing distance from two sides and their included angle, solving a triangle from two known angles and one measured side, or confirming area and perimeter before transferring dimensions into a drawing or spreadsheet.
How to calculate. The calculator opens with a complete demonstration: Given is set to Three sides (SSS), Length unit is centimeters, and Side a, Side b, and Side c are 5, 7, and 8. The first solution and its Excel workbook are available immediately. To solve your own triangle:
- Choose the matching option in Given: Three sides (SSS), Two sides + included angle (SAS), Two angles + included side (ASA), or Two angles + adjacent side (AAS).
- Select one Length unit. Changing it converts every entered side value so the represented physical triangle stays the same.
- Replace the visible side and angle values. Results update live as soon as the active data set is complete and valid.
- Read the Area first, then confirm the solved sides, angles, Perimeter, Semiperimeter, formula note, and Solution details table.
- Select Download Excel to build a fresh workbook from the current validated model. Reset clears the demonstration and all result state; Excel export then remains disabled until a complete valid data set is entered again.
Input guide. Given is required and determines which fields are active. SSS requires Side a, Side b, and Side c; each must be a positive decimal and must satisfy the triangle inequality. SAS requires positive Side a and Side b plus Angle γ, the angle between those sides. ASA uses Angle β, Angle γ, and Side a, because side a lies between β and γ. AAS uses Angle β, Angle γ, and adjacent Side b. Length unit is required and accepts mm, cm, m, km, in, ft, or yd. Side fields accept standard decimal notation with optional correctly placed comma grouping, such as 1,250.5; decimal commas and scientific notation are rejected to prevent silent reinterpretation. Each side must be greater than 0 and no greater than one trillion selected units. Angle β and Angle γ are required only in the modes that show them; enter degrees greater than 0 and less than 180, and for ASA or AAS their sum must be less than 180. A realistic SAS example is Side a = 4, Side b = 5, and Angle γ = 40°. A common mistake is pairing a side with the wrong opposite angle: side a is always opposite α, side b opposite β, and side c opposite γ.
Output guide. Area is the enclosed surface in square selected units. Side a, Side b, and Side c are exact model lengths displayed with up to six decimals. Angle α, Angle β, and Angle γ are interior angles shown in degrees; Angle sum is an identity check and should read 180°. Perimeter is a + b + c, while Semiperimeter is half that value and is the s term in Heron's formula. Mode reports the active data pattern. Validity identifies whether the solved triangle is acute, obtuse, or on the right-angle boundary. The formula note states which laws produced the solution. In the Solution details table, Quantity names the model field, Value gives its formatted result, and Interpretation explains its geometric role. A zero or negative area is never a valid solved triangle; it indicates missing data, a degenerate angle combination, or sides that cannot close.
Worked example. With the opening SSS values a = 5 cm, b = 7 cm, and c = 8 cm, the perimeter is 20 cm and the semiperimeter is s = 10 cm. Heron's formula gives area = √(10 × 5 × 3 × 2) = √300 = 17.320508 cm². The law of cosines then gives α = 38.2132°, β = 60°, and γ = 81.7868°, which total 180°. These values match the first rendered results and the startup Excel checkpoints. For a deeper derivation of the area step, see Wolfram MathWorld's explanation of Heron's formula.
Learn more. SSS and SAS are naturally handled by the law of cosines, while ASA and AAS use the angle-sum identity followed by the law of sines. OpenStax's section on the law of cosines for SSS and SAS triangles explains why the included angle matters. The calculator deliberately omits SSA because it can be ambiguous rather than uniquely solvable.
Formulas and validity checks
For SSS, the calculator first verifies a + b > c, a + c > b, and b + c > a. It then uses s = (a + b + c) / 2 and area = √(s(s – a)(s – b)(s – c)). Each angle is recovered with a clamped inverse-cosine expression so floating-point noise cannot push a theoretically valid cosine slightly outside – 1 to 1.
For SAS, c² = a² + b² – 2ab cos γ and area = ½ab sin γ. For ASA and AAS, the unknown third angle equals 180° minus the two known angles. The remaining sides follow from a / sin α = b / sin β = c / sin γ. These equations are exact geometric identities; only the displayed text is rounded.