SAS Triangle Calculator

How to use the SAS triangle calculator

What this calculator does

This calculator solves a triangle when you know two adjacent side lengths and the angle included between them – the side-angle-side, or SAS, data set. From Side a, Side b, and Included angle γ, it calculates the opposite Side c, the remaining Angle α and Angle β, the Area, Perimeter, semiperimeter, a useful height, and geometric classifications. SAS determines one unique nondegenerate triangle when both sides are positive and the included angle is strictly between 0 and 180 degrees. The calculator does not solve the ambiguous SSA case, where the known angle is not between the two known sides.

When to use it

Use this tool to check a trigonometry exercise, determine a diagonal or separation that cannot be measured directly, estimate the area enclosed by two measured segments, or verify dimensions in surveying, fabrication, carpentry, CAD, and general geometry work. It is also useful for comparing how a triangle changes when one side or the included angle changes while the other known values stay fixed.

How to calculate

1. The calculator opens with a complete demonstration: a = 4 cm, b = 3 cm, and γ = 42°. The results and a validated example Excel workbook are available immediately.
2. Replace Side a and Side b with positive decimal lengths. Standard decimal notation is accepted; commas may be used only as three-digit thousands separators.
3. Enter the Included angle γ. Choose Degrees (°) or Radians (rad) with the Angle unit control. The angle must remain greater than zero and less than a straight angle.
4. Use Length unit to convert both known sides and every length-based output together. The physical triangle is preserved during unit conversion.
5. Read the primary missing side, the result cards, geometry checks, scaled diagram, and detail table. Select Download Excel to export the current validated model as a real .xlsx workbook.
6. Reset clears the demonstration and all calculated content. Download Excel is then disabled until a complete valid SAS data set is entered again.

Input guide

Side a is a required positive length adjacent to γ and opposite α. Enter a plain decimal such as 4 or 4.25 in the selected Length unit. A larger a generally increases area and perimeter and can change every solved angle. Do not enter a unit symbol, scientific notation, zero, or a negative length in the field.

Side b is the second required positive length adjacent to γ and opposite β. A realistic example is 3 cm. Its accepted format and limits match Side a. A common mistake is to enter a side that is opposite γ; in SAS notation, the known sides must meet at γ.

Included angle γ is required. In degree mode, enter a value strictly between 0 and 180, such as 42. In radian mode, enter a value strictly between 0 and π, such as 0.733038. Increasing γ usually opens the triangle, increasing side c and area until the area peaks at 90°, after which area falls while c continues toward a + b. The most important interpretation error is using an angle that is not between a and b.

Length unit is required and may be mm, cm, m, km, in, ft, or yd. Changing it converts the existing side entries and all length, area, perimeter, and height outputs. Angle unit is required and switches between degrees and radians; it converts the current γ value and all angle outputs rather than merely relabeling them. NIST identifies the radian as the coherent SI unit for plane angle.

Output guide

Missing side c is the length opposite γ and is the primary solved result. It is an exact geometric consequence of the three inputs, subject only to floating-point and displayed rounding. Angle α is opposite a, and Angle β is opposite b. Their displayed sum with γ should be 180° or π radians. A near-zero angle indicates a very narrow triangle; no valid angle can be negative.

Area measures the enclosed two-dimensional region in square units and is driven by both side lengths and sin(γ). An area approaching zero means the triangle is approaching a degenerate straight-line shape. Perimeter is a + b + c, while Semiperimeter is half that value. Height to side b is the perpendicular height when b is treated as the base. The Longest side and Largest angle checks should correspond because the greatest side in any triangle lies opposite the greatest angle.

The summary pills identify the angle class – acute, right, or obtuse – and the side class – equilateral, isosceles, or scalene. The scaled diagram uses the same model values and preserves side ratios and angle shape, but it is resized for the available card. The details table lists each quantity, its calculation path, and the rounded current value.

Worked example

For the startup values a = 4 cm, b = 3 cm, and γ = 42°, the law of cosines gives c = √(4² + 3² – 2 × 4 × 3 × cos 42°) = 2.6767 cm. Solving the other angles gives α = 89.4130° and β = 48.5870°, so the angle sum is 180.0000°. The area is ½ × 4 × 3 × sin 42° = 4.0148 cm², and the perimeter is 4 + 3 + 2.6767 = 9.6767 cm. OpenStax provides a detailed derivation and examples for the law of cosines for non-right triangles.

Learn more

After the third side is known, the remaining angles can be checked with the law of sines; see the OpenStax law of sines and oblique-triangle area guide. For numerical work, keep more digits than you plan to report and round only the final displayed result. Extremely small or nearly straight included angles can make a mathematically valid triangle sensitive to tiny measurement errors.