Oblique Triangle Solver (SAS) Calculator

Enter two known sides and the included angle (SAS) to calculate the third side, remaining angles, area, and perimeter of the triangle.

Created by Luis GonzalezLast updated: May 9, 2026

How to Use This Calculator

1
Input Side a
Enter the length of the first known side, adjacent to the included angle.
2
Input Side b
Provide the length of the second known side, also adjacent to the included angle. Ensure units are consistent with Side a.
3
Input Included Angle C (°)
Enter the measure of the angle formed *between* Side a and Side b, in degrees. This angle must be between 0° and 180°.
4
Review Your Results
The calculator will instantly display the length of the unknown side (Side c), the remaining angles (Angle A, Angle B), area, perimeter, and aspect ratio.

Example Calculation

A carpenter is cutting a triangular piece of wood and knows two side lengths and the angle between them.

Side a

Side b

Included Angle C (°)

Results

6.2209

Tips

Angle C Range

The included angle (Angle C) must be strictly between 0° and 180°. An angle of 0° or 180° would result in a degenerate triangle (a straight line), which is not a valid geometric shape for this calculation.

Law of Cosines Precision

The Law of Cosines involves squaring side lengths and using cosine values. For accurate results, especially with non-integer inputs, ensure your measurements are as precise as possible (e.g., to two decimal places for lengths) to minimize cumulative rounding errors.

Unit Consistency for Sides

Always use the same unit for both Side a and Side b. The calculated Side c, perimeter, and related length-based outputs will then be in that same unit, preventing conversion errors.

The Oblique Triangle Solver (SAS) Calculator is a vital tool for professionals and students alike, enabling the precise determination of unknown dimensions in a triangle when two sides and their included angle are known. This calculator quickly computes the length of the third side, the measures of the two remaining angles, the area, and the perimeter. For structural engineers, calculating the third side with an accuracy of 0.01 meters can be the difference between a stable design and a compromised one in 2025 projects.

The Law of Cosines for Solving SAS Triangles

The fundamental principle for solving SAS (Side-Angle-Side) triangles is the Law of Cosines. This powerful law is a generalization of the Pythagorean theorem, extending its applicability to all triangles, whether right-angled or oblique. It is particularly useful when you know two sides and the angle between them, as it directly provides the length of the third unknown side.

The Law of Cosines states:

c^2 = a^2 + b^2 - 2ab × cos(C)

Where a and b are the lengths of the two known sides, C is the measure of the angle included between sides a and b, and c is the length of the side opposite angle C. Once c is found, the Law of Sines can then be used to find the remaining angles.

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Constructing a Triangular Garden Bed

A homeowner is planning to build a triangular raised garden bed. They have two sides measured at 7 feet and 10 feet, with the angle between them (the included angle C) set at 38°. They need to find the length of the third side and the other angles to cut the remaining materials.

Here's the step-by-step calculation:

Calculate Side c: Using the Law of Cosines: c² = 7² + 10² - (2 × 7 × 10 × cos(38°)). c² = 49 + 100 - (140 × 0.7880) = 149 - 110.32 = 38.68. So, c = √38.68 ≈ 6.2209 feet.
Calculate Angle A: Now that we have Side c, we can use the Law of Sines: sin(A) / a = sin(C) / c. sin(A) / 7 = sin(38°) / 6.2209. sin(A) = (7 × sin(38°)) / 6.2209 = (7 × 0.6157) / 6.2209 ≈ 0.6928. A = arcsin(0.6928) ≈ 43.79°.
Calculate Angle B: The sum of angles in a triangle is 180°. So, B = 180° - 38° - 43.79° = 98.21°.
Area and Perimeter: The perimeter is 7 + 10 + 6.2209 = 23.2209 feet. The area, using Heron's formula or (1/2)ab sin(C), is approximately 20.9633 square feet.

The length of Side c is approximately 6.2209 feet.

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Alternative Methods for Solving SAS Triangles

While the Law of Cosines is the most direct and efficient method for solving SAS triangles, alternative approaches exist, primarily involving the decomposition of the oblique triangle. One common method is to drop an altitude from one of the known vertices to the opposite side (or its extension), thereby creating two right-angled triangles. This allows for the use of basic right-triangle trigonometry (SOH CAH TOA) and the Pythagorean theorem to solve for unknown lengths and angles.

For example, given sides a, b, and included angle C:

Drop a perpendicular from vertex B to side a (or its extension), creating a height h and two segments on side a.
Use h = c × sin(A) (where A is one of the known angles) and the definition of cosine to find segments.
Apply the Pythagorean theorem to the two right triangles.

This method, while effective, often involves more steps and intermediate calculations than the direct application of the Law of Cosines, which elegantly combines these geometric relationships into a single formula. The Law of Cosines is particularly advantageous when only the final unknown side is needed, whereas decomposing the triangle might be useful if the altitude itself or the segments on the base are of interest.

Precision in Engineering and Design

In engineering disciplines, such as mechanical design or civil engineering, the accuracy of SAS triangle solutions is paramount. When designing components like linkages, robotic arms, or bridge sections, even small deviations in calculated side lengths or angles can lead to significant structural flaws or functional errors. For instance, in robotics, the forward kinematics of a multi-jointed arm often involve a chain of SAS calculations to determine the precise position of the end-effector. Engineers typically work within tolerances of fractions of a millimeter or degrees, and the robustness of their designs depends heavily on the accuracy of these trigonometric computations. The Law of Cosines provides a mathematically sound basis for these critical precision requirements.

Frequently Asked Questions

What is an SAS triangle?

An SAS (Side-Angle-Side) triangle is a triangle where the lengths of two sides and the measure of the angle *included* between those two sides are known. This specific combination of information is one of the congruence criteria, meaning it uniquely determines the shape and size of the triangle. Knowing these three elements allows for the calculation of all other unknown properties, such as the third side, the remaining two angles, and the triangle's area.

Why is the Law of Cosines essential for SAS triangles?

The Law of Cosines is essential for SAS triangles because it directly relates the three sides of a triangle to one of its angles. When you know two sides and the included angle, the Law of Cosines provides a direct formula to find the length of the third unknown side. This law is a generalization of the Pythagorean theorem and is indispensable for solving non-right triangles where the Law of Sines alone would be insufficient without an opposite angle-side pair.

In which real-world scenarios are SAS triangles commonly encountered?

SAS triangles are frequently encountered in engineering, construction, and navigation. For example, a construction worker might use an SAS calculation to determine the length of a diagonal brace needed for a rectangular frame after measuring two adjacent sides and the angle between them if it's not a perfect 90-degree corner. Similarly, in robotics, determining the end-effector position often involves SAS triangle solutions based on arm segment lengths and joint angles.