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Triangle Calculator – Law of Sines & Cosines

Fully solve any oblique or general triangle from 3 given parameters (SSS, SAS, ASA, SSA). Calculates all angles, sides, heights, area (Heron & trigonometric), incircle, circumcircle, and full step-by-step solutions.

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Dreiecks-Beispiele (Presets)

Eingabewerte für SSS

Fläche A24.00 cm²
Umfang U24.00 cm
Inkreis r / Umkreis Rr=2.00 | R=5.00
Winkel (α / β / γ)36.9° / 53.1° / 90.0°

Dreiecks-Visualisierung (2D)

c = 10.0a = 6.0b = 8.0A (α=36.9°)B (β=53.1°)C (γ=90.0°)
Winkel-Typ: rightSeiten-Typ: scaleneFläche: 24.00 cm²Umfang: 24.00 cm

📐Schritt-für-Schritt Herleitung (Sinus- & Kosinussatz)

Gegeben: a=6,b=8,c=10 (SSS-Satz)\text{\textbf{Gegeben: }} a = 6, b = 8, c = 10 \text{ (SSS-Satz)}
1. Winkel α u¨ber den Kosinussatz:\text{\textbf{1. Winkel }} \alpha \text{ über den Kosinussatz:}
cos⁡(α)=b2+c2−a22bc=82+102−622⋅8⋅10=0.8000  ⟹  α=36.87∘\cos(\alpha) = \frac{b^2 + c^2 - a^2}{2bc} = \frac{8^2 + 10^2 - 6^2}{2 \cdot 8 \cdot 10} = 0.8000 \implies \alpha = 36.87^\circ
2. Winkel β u¨ber den Kosinussatz:\text{\textbf{2. Winkel }} \beta \text{ über den Kosinussatz:}
cos⁡(β)=a2+c2−b22ac=62+102−822⋅6⋅10=0.6000  ⟹  β=53.13∘\cos(\beta) = \frac{a^2 + c^2 - b^2}{2ac} = \frac{6^2 + 10^2 - 8^2}{2 \cdot 6 \cdot 10} = 0.6000 \implies \beta = 53.13^\circ
3. Winkel γ u¨ber die Winkelsumme: γ=180∘−α−β=90.00∘\text{\textbf{3. Winkel }} \gamma \text{ über die Winkelsumme: } \gamma = 180^\circ - \alpha - \beta = 90.00^\circ
4. Fla¨cheninhalt und Umfang:\text{\textbf{4. Flächeninhalt und Umfang:}}
U=a+b+c=6.00+8.00+10.00=24.0000U = a + b + c = 6.00 + 8.00 + 10.00 = 24.0000
A=s(s−a)(s−b)(s−c)=24.0000(mit s=U2=12.00)A = \sqrt{s(s-a)(s-b)(s-c)} = 24.0000 \quad (\text{mit } s = \frac{U}{2} = 12.00)
Ho¨hen: ha=8.0000,  hb=6.0000,  hc=4.8000\text{Höhen: } h_a = 8.0000, \; h_b = 6.0000, \; h_c = 4.8000
Radien: Inkreis r=As=2.0000,  Umkreis R=a2sin⁡(α)=5.0000\text{Radien: Inkreis } r = \frac{A}{s} = 2.0000, \; \text{Umkreis } R = \frac{a}{2\sin(\alpha)} = 5.0000

Historical Origin & The Challenge Solved

Why was this mathematics developed, where does it come from, and what real-world problem did it solve?

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Epoche & ZeitalterHellenistic Era & Islamic Astronomy (c. 150 BCE – 1464 CE)
Pioniere: Hipparchus of Nicaea, Al-Battani, Regiomontanus (Johannes Müller)

1. Inaccessible Peaks & Open Sea Navigation

How do you determine the distance to an enemy warship at sea or the elevation of an unclimbable mountain without physically pacing the distance? Classical geometry failed when target points were unreachable.

2. Triangulation, Law of Sines & Law of Cosines

Regiomontanus published the foundational work "De Triangulis Omnimodis" in 1464. Trigonometry proved that measuring one baseline on the ground and two angles sufficed to calculate all sides and angles of any triangle.

3. Geodesy, Satellite Cartography & Robotics

Global topography mapping and inverse kinematics in multi-joint robotic arms are solved using trigonometric triangle solvers.

„He who masters trigonometry can measure heaven and earth without leaving his seat.“ — Regiomontanus

Real-World Applications of Triangle Trigonometry

Discover how the law of sines, cosines, and triangulation are applied in geodesy, satellite navigation, and structural design.

Geodesy & Surveying

Triangulation & Land Surveying

Determining inaccessible mountain heights and river distances using baseline sights and theodolite angle readings.

Praxis-Beispiel:

Measuring distance to a landmark from two baseline points using the law of sines.

Aviation & Marine Navigation

Dead Reckoning & Wind Vector Correction

Correcting aircraft drift caused by crosswinds using vector triangles and the law of cosines.

Praxis-Beispiel:

Computing true heading and ground speed adjustments in adverse weather conditions.

Architecture & Civil Engineering

Truss Frameworks & Roof Framing

Structural load analysis and rafter geometry in triangular truss bridges and pitched roofs.

Praxis-Beispiel:

Calculating exact rafter lengths and pitch angles for asymmetrical timber roof trusses.

Guide: General Triangles, Law of Sines & Cosines

Unlike right triangles, oblique triangles have no 90° angle. The Law of Sines and Law of Cosines allow complete calculation of all unknown sides and angles.

Law of Sines

Side lengths are proportional to the sines of their opposite angles: a / sin(α) = b / sin(β) = c / sin(γ) = 2R.

Law of Cosines

Generalization of Pythagorean theorem for any triangle: c² = a² + b² - 2ab · cos(γ).

Trigonometric Area Formula

Calculate area using two sides and their included angle: A = ½ · a · b · sin(γ).

Heron’s Formula

Compute area strictly from 3 side lengths with semi-perimeter s = (a+b+c)/2: A = √(s(s-a)(s-b)(s-c)).

Frequently Asked Questions about Triangle Calculations

When do I use the Law of Sines versus the Law of Cosines?▼

Use the Law of Cosines when given three sides (SSS) or two sides and the included angle (SAS). Use the Law of Sines when at least one opposite side-angle pair is known (ASA, AAS, SSA).

What is the Triangle Inequality Theorem?▼

The sum of any two side lengths must always be strictly greater than the third side (a + b > c). Otherwise, no triangle can form.

What is the difference between incircle and circumcircle?▼

The circumcircle passes through all three triangle vertices (center: perpendicular bisectors). The incircle is tangent to all three inner sides (center: angle bisectors).

How do you calculate triangle heights (altitudes)?▼

Heights derive directly from area A: h_a = 2A / a, h_b = 2A / b, and h_c = 2A / c.