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Sas Triangle Calculator

Calculate the missing side and area of any triangle from two known sides and their included angle using the Law of Cosines and SAS area formula.

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What Is SAS Triangle Calculation?

SAS — short for Side-Angle-Side — describes a triangle defined by two known side lengths and the angle between them. Given this information, every property of the triangle can be determined uniquely. The SAS triangle calculator automates two foundational trigonometric formulas: the Law of Cosines for computing the missing third side, and the SAS area formula for computing the triangle's enclosed area.

The Law of Cosines: Core Formula

When sides a and b and the included angle C are known, the missing side c is found using:

c = √(a² + b² − 2ab · cos(C))

As documented by Texas A&M University Math 150, Section 8.5 — Law of Cosines, this equation generalizes the Pythagorean theorem to oblique triangles. When C = 90°, cos(90°) = 0, so the correction term 2ab·cos(C) vanishes, reducing the formula to the classic c² = a² + b². For acute angles, the positive cosine term shortens c relative to the Pythagorean prediction; for obtuse angles, the negative cosine term lengthens it.

Understanding the Variables

  • Side a — The first known side, measured in any consistent unit such as meters, feet, or centimeters.
  • Side b — The second known side, in the same unit as side a.
  • Angle C — The included angle in degrees, positioned directly between sides a and b. This is the only angle needed for the SAS calculation.
  • Side c — The computed side opposite angle C, produced by the Law of Cosines.

The SAS Area Formula

With two sides and their included angle, the triangle's area is:

Area = ½ · a · b · sin(C)

Montana's Tipi Geometry & Trigonometry curriculum for Grades 10–12 demonstrates this formula geometrically: the product b·sin(C) equals the perpendicular height of the triangle when side a serves as the base. This makes the SAS formula equivalent to Area = ½ · base · height, without requiring the height to be measured directly.

Geometric Derivation of the Law of Cosines

Place side a along the horizontal axis with one endpoint at the origin. Extend side b from the origin at angle C, so its far endpoint lands at coordinates (b·cos(C), b·sin(C)). Side c connects the point (a, 0) to (b·cos(C), b·sin(C)). Applying the distance formula gives c² = (a − b·cos(C))² + (b·sin(C))². Expanding and applying the identity sin²(C) + cos²(C) = 1 yields c² = a² + b² − 2ab·cos(C), confirming the Law of Cosines from first principles.

Worked Example: Land Survey

A surveyor measures two boundary fences of a triangular plot: a = 120 m and b = 95 m, with an included angle of C = 58°.

  • Missing side: c = √(120² + 95² − 2 · 120 · 95 · cos(58°)) = √(14400 + 9025 − 12082) = √11343 ≈ 106.5 m
  • Area: Area = 0.5 · 120 · 95 · sin(58°) = 5700 · 0.848 ≈ 4833 m²

These two results fully characterize the triangle's size and shape. From side c, the remaining angles A and B can be found using the Law of Sines or a second application of the Law of Cosines, with all three angles summing to exactly 180°.

Practical Applications of SAS Triangles

SAS calculations appear across many professional fields: computing roof diagonal spans from two rafter lengths and a ridge angle, determining GPS distances between two waypoints through a known vertex, mapping agricultural field boundaries, designing structural bracing in civil and mechanical engineering, and solving navigation problems in aeronautics and maritime routing. Architects use SAS to calculate diagonal wall lengths and floor-to-ceiling dimensions in building designs. Manufacturing engineers employ SAS calculations when designing mechanical linkages and robotic arms. In surveying and land development, SAS triangulation is essential for determining property line lengths and areas from measured field data. Any scenario where two sides and the enclosed angle are directly measurable — but the third side is not — calls for the SAS method and the Law of Cosines. The calculator's automation eliminates tedious manual computations, reducing errors and accelerating project timelines across these diverse technical disciplines.

Reference

Frequently asked questions

What does SAS mean in triangle geometry?
SAS stands for Side-Angle-Side, one of the fundamental triangle congruence postulates in Euclidean geometry. It specifies that two side lengths and the angle directly between them — called the included angle — are known. This combination uniquely defines the triangle: no other triangle can exist with those exact three measurements, making the calculation precise and unambiguous every time.
How does the Law of Cosines differ from the Pythagorean theorem?
The Pythagorean theorem (c² = a² + b²) applies only to right triangles where one angle is exactly 90°. The Law of Cosines (c² = a² + b² − 2ab·cos(C)) works for any triangle regardless of its angles. When C equals 90°, cos(90°) = 0, so the correction term disappears and the formula reduces exactly to the Pythagorean theorem, confirming the Law of Cosines as the universal generalization that subsumes it.
What is the formula for the area of a triangle given two sides and the included angle?
The SAS area formula is: Area = (1/2) · a · b · sin(C), where a and b are the two known sides and C is the included angle in degrees. For example, a triangle with sides of 10 cm and 14 cm meeting at 45° has an area of 0.5 × 10 × 14 × sin(45°) ≈ 49.5 square centimeters. This approach eliminates the need to measure the triangle's perpendicular height directly.
Can the SAS triangle calculator handle obtuse angles greater than 90 degrees?
Yes. Both the Law of Cosines and the SAS area formula work for any angle strictly between 0° and 180°. For an obtuse included angle such as C = 120°, cos(120°) = −0.5, making the 2ab·cos(C) correction term negative, which correctly produces a longer third side. The sine function remains positive throughout the 0°–180° range, so the area formula is fully valid for obtuse triangles as well.
What units should be used when entering sides and the included angle?
Side lengths can be entered in any unit — meters, feet, centimeters, or inches — provided both sides use the same unit. The computed third side c will be expressed in that same unit, and the area will be in the corresponding square units. The included angle C must be entered in degrees. No manual conversion to radians is necessary; the calculator applies the trigonometric functions internally using the correct conversion.
How are the remaining angles A and B determined after computing the missing side?
Once side c is known from the Law of Cosines, angles A and B can be found using either the Law of Sines — sin(A)/a = sin(C)/c — or a second application of the Law of Cosines: cos(A) = (b² + c² − a²) / (2bc). As a verification step, angles A, B, and C must sum to exactly 180°. The SAS method fully determines the triangle, making all six elements — three sides and three angles — computable from the original three inputs.