🎯 Triangle Calculators Solver

Law of Cosines Calculator

Solve SAS and SSS triangles instantly using the Law of Cosines (c² = a² + b² - 2ab·cos C) with complete step-by-step solutions.

Known Triangle Parameters (SAS)

Enter values to calculate instant result

Side (a) ℹ️ Info

Side (b) ℹ️ Info

Included Angle C (Between a and b) (°) ℹ️ Info

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Calculation Reference Table

Benchmark values and quick reference intervals

Top 10 Reference Values

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How to Use & Educational Guide

In-depth guide for Law of Cosines Calculator (c² = a² + b² - 2ab cos C)

Educational Overview

Solving SAS Triangles: Core Principles, Law of Cosines, and Geometric Constraint

In planar Euclidean geometry, a Side-Angle-Side (SAS) configuration describes a triangle where two adjacent boundary side lengths ($a$ and $b$) and the specific angle strictly enclosed between them ($\angle C$, or $\gamma$) are known. By Euclid’s fundamental congruence theorem (Book I, Proposition 4), knowing two sides and their included angle creates a completely rigid geometric framework: it fixes the exact spatial coordinates of all three vertices and uniquely determines the entire triangle.

Unlike configurations involving non-included angles (such as SSA, which frequently produces ambiguous multi-triangle solutions), an SAS triangle produces exactly one unique solution. Because you do not possess a matched side-and-opposite-angle pair, the elementary Law of Sines cannot initiate solving. Instead, the generalized Pythagorean relationship—the Law of Cosines—serves as the primary mathematical key to bridge the two known arms across the vertex hinge, calculating the unknown opposing side ($c$) directly before determining remaining angles and surface area.

SAS Parameter Configuration

c² = a² + b² - 2ab·cos(C) | Angle C enclosed between a & b

The Geometric Hinge Principle

Wider included angle C opens the opposing span c

How to Use the SAS Triangle Calculator Step-by-Step

This tool is specifically engineered for two-arm angle measurements. By providing the two known side lengths and the angle between them, you compute all remaining dimensions instantly:

Enter Side (a): Type the linear length of the first known side into the "Side (a)" box.
Enter Side (b): Type the linear length of the second known side into the "Side (b)" box. Verify that both side inputs are measured in the same unit system (e.g., centimeters, meters, inches, or feet).
Enter Included Angle C (°): Type the vertex angle formed between side $a$ and side $b$ in degrees. For any valid Euclidean planar triangle, this value must be greater than $0^\circ$ and strictly less than $180^\circ$.
Instant Live Calculation: In 0ms, the calculator evaluates the unknown opposing side ($c$), solves both remaining interior angles ($\angle A$ and $\angle B$), computes the exact surface area and total perimeter, and updates an interactive, proportionally scaled vector drawing. Use the Reset button at any time to clear all fields.

The Mathematical Formulation: From Side c to Angles and Area

Solving an SAS triangle follows a clean, three-stage sequential mathematical pipeline:

Step 1: Solve Opposite Side (c)

$$c = \sqrt{a^2 + b^2 - 2ab\cos C}$$

Applies the Law of Cosines across included angle $C$.

Step 2: Solve Remaining Angles

$$\cos A = \frac{b^2 + c^2 - a^2}{2bc}$$

$$B = 180^\circ - (A + C)$$

Step 3: Direct Surface Area

$$\text{Area} = \frac{1}{2}ab\sin C$$

Computes planar area directly without finding height.

Reference Solutions Across Standard SAS Scenarios

This reference table illustrates solved opposite sides ($c$), interior angles ($\angle A, \angle B$), and surface areas across varied SAS inputs:

Side (a)	Side (b)	Angle C (°)	Opposite Side (c)	Angle A (°)	Angle B (°)	Area
8.00	11.00	42.00°	7.36	46.68°	91.32°	29.44
15.00	20.00	90.00°	25.00	36.87°	53.13°	150.00
14.00	19.00	115.00°	27.96	27.02°	37.98°	120.54
10.00	10.00	60.00°	10.00	60.00°	60.00°	43.30
6.50	13.20	34.50°	8.71	25.00°	120.50°	24.28

Worked Practical Examples with Varied Scenarios

Example 1: Acute Triangle Solving (Side a = 9 m, Side b = 13 m, Angle C = 52°)

• Step 1 (Find Side c): $c^2 = 9^2 + 13^2 - 2(9)(13)\cos(52^\circ)$.

• Step 2 (Evaluate Terms): $81 + 169 - (234 \times 0.61566) = 250 - 144.06 = \mathbf{105.94}$.

• Step 3 (Extract Root): $c = \sqrt{105.94} \approx \mathbf{10.29\text{ m}}$.

• Step 4 (Find Angle A): $\cos A = \frac{13^2 + 10.29^2 - 9^2}{2(13)(10.29)} = \frac{169 + 105.88 - 81}{267.54} = \frac{193.88}{267.54} \approx 0.7247 \implies A \approx \mathbf{43.56^\circ}$.

• Step 5 (Find Angle B & Area): $B = 180^\circ - (52^\circ + 43.56^\circ) = \mathbf{84.44^\circ}$. $\text{Area} = \frac{1}{2}(9)(13)\sin(52^\circ) \approx \mathbf{46.10\text{ m}^2}$.

👉 Results: c ≈ 10.29 m, A ≈ 43.56°, B ≈ 84.44°, Area ≈ 46.10 m²

Example 2: Obtuse Surveying Triangle (Side a = 17 ft, Side b = 23 ft, Angle C = 124°)

• Step 1 (Cosine Value): $\cos(124^\circ) \approx -0.55919$ (Notice negative sign for angle $> 90^\circ$).

• Step 2 (Apply Law of Cosines): $c^2 = 17^2 + 23^2 - [2(17)(23)(-0.55919)] = 289 + 529 - [-437.29] = 818 + 437.29 = \mathbf{1255.29}$.

• Step 3 (Extract Opposite Side): $c = \sqrt{1255.29} \approx \mathbf{35.43\text{ ft}}$.

• Step 4 (Surface Area): $\text{Area} = \frac{1}{2}(17)(23)\sin(124^\circ) \approx 195.5 \times 0.82904 = \mathbf{162.08\text{ sq ft}}$.

👉 Results: Opposing Side c ≈ 35.43 ft, Surface Area ≈ 162.08 sq ft

Practical Applications Across Industries

🗺️ Land Surveying & Inaccessible Lines

Surveyors calculate the straight-line boundary distance between two property corners separated by dense woods or rivers by standing at an offset point and measuring two sightlines and the angle between them.

✈️ Aviation & Dead Reckoning

Flight computers determine aircraft displacement and ground distance when an aircraft changes heading by a set course angle after flying along two separate straight-line flight legs.

🤖 Robotic Arm Linkages

Forward kinematics controllers compute the exact Cartesian coordinate of a robotic gripper by evaluating the known segment arm lengths ($a, b$) and the elbow servo hinge angle ($C$).

🏗️ Structural Roof Trusses

Carpenters and structural engineers compute the horizontal bottom chord tie-beam length ($c$) required to support two pitched rafters ($a, b$) joined at a specific ridge apex angle ($C$).

⛵ Marine Navigation & Tacking

Sailors calculate net distance made good toward a destination after sailing two successive tacking legs at a known tack angle against prevailing headwinds.

🎮 Game Physics & Trajectories

Game physics engines compute line-of-sight clearance and AI field-of-view interception vectors between two game entities moving along diverging velocity vectors.

Common Pitfalls & Mistakes to Avoid

Entering a Non-Included Angle: The angle entered must be the vertex angle enclosed between side $a$ and side $b$. If your known angle lies opposite to side $a$ or side $b$ (SSA configuration), this is not an SAS triangle and can trigger the ambiguous two-triangle case.
Neglecting Negative Signs for Obtuse Angles: When angle $C > 90^\circ$, its cosine is negative (e.g., $\cos(120^\circ) = -0.5$). Subtracting a negative quantity results in addition: $-2ab(-0.5) = +ab$. Forgetting this negative sign produces an opposite side ($c$) that is far too short.
PEMDAS Order of Operations Failure: A frequent manual arithmetic mistake is calculating $(a^2 + b^2 - 2ab) \times \cos C$. Multiplication has higher precedence: calculate the product $2ab\cos C$ first, then subtract that value from $(a^2 + b^2)$.
Angle Range Boundary Violations: In Euclidean plane geometry, an interior angle must be strictly between $0^\circ$ and $180^\circ$. Entering $0^\circ$ collapses the triangle into a 1D line segment, while $180^\circ$ forms a straight line with zero surface area.

Mathematical accuracy verified by AptCalc Engine

Frequently Asked Questions (FAQs)

Answers to common questions about this calculator

What does SAS mean in geometry?

SAS stands for Side-Angle-Side. It describes a geometric condition where you know the lengths of two sides of a triangle and the measure of the angle enclosed strictly between them.

Can an SAS triangle ever have more than one solution?

No. By Euclidean congruence principles, an SAS configuration is completely rigid and unique. Unlike the SSA condition (which can produce zero, one, or two triangles), an SAS triangle always yields exactly one valid triangle.

Why can’t you use the Law of Sines to start solving an SAS triangle?

The Law of Sines requires at least one complete ratio of a side and its direct opposite angle (such as a and angle A). In an SAS triangle, you only know side a, side b, and angle C, meaning you do not have a single complete side-to-opposite-angle pair. Therefore, you must use the Law of Cosines first to find side c.

How do you find the area of an SAS triangle without knowing the height?

You can compute the area directly using the SAS trigonometric area formula: Area = ½ × a × b × sin(C). This eliminates the need to construct or calculate an internal perpendicular altitude.

What happens to the formula if included angle C is exactly 90 degrees?

If angle C = 90°, cos(90°) = 0, which completely cancels out the -2ab·cos(C) term. The Law of Cosines simplifies directly to c² = a² + b², matching the classical Pythagorean theorem for right triangles.