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Oblique Triangle Calculator

Solve any oblique triangle by entering SSS, SAS, ASA, or AAS values. Instantly computes all sides, angles, and area using the Law of Sines and Law of Cosines.

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Triangle Area—sq units

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What Is an Oblique Triangle?

An oblique triangle is any triangle that contains no right (90°) angle. Unlike right triangles, oblique triangles require the Law of Sines, the Law of Cosines, or the trigonometric area formula to resolve unknown sides, angles, and area. The oblique triangle calculator selects the correct formula automatically based on which three elements are provided: SSS, SAS, ASA, or AAS.

The Three Governing Formulas

1. Law of Sines

The Law of Sines states that each side divided by the sine of its opposite angle produces a constant ratio throughout the triangle:

a / sin(A) = b / sin(B) = c / sin(C)

This relationship applies directly to the ASA case (two angles and the included side) and the AAS case (two angles and a non-included side). Because all interior angles in any triangle sum to 180°, knowing two angles immediately determines the third, making the Law of Sines the efficient choice for angle-rich inputs.

2. Law of Cosines

The Law of Cosines extends the Pythagorean theorem to triangles of any shape:

c² = a² + b² − 2ab · cos(C)

Equivalent forms solve for sides a and b by substituting the respective opposite angles. Setting C = 90° reduces cos(C) to zero, collapsing the expression to the classic Pythagorean identity c² = a² + b². The Law of Cosines is essential for the SAS case (two sides with the included angle) and the SSS case (all three sides known).

3. Area Formula

The trigonometric area formula computes triangle area from two sides and their included angle, eliminating the need to measure height:

Area = ½ · a · b · sin(C)

For example, a triangle with a = 7 m, b = 10 m, and included angle C = 45° has an area of ½ × 7 × 10 × sin(45°) ≈ ½ × 70 × 0.7071 ≈ 24.75 m². This formula derives from the standard base-times-height formula by expressing height as b · sin(C).

Solving the Four Input Cases

SSS — Three Sides Known

Apply the Law of Cosines to the largest angle first to minimize rounding error. Given sides a = 5, b = 7, c = 9: cos(C) = (a² + b² − c²) / (2ab) = (25 + 49 − 81) / 70 = −7/70 ≈ −0.1, so C ≈ 95.74°. The Law of Sines then yields A and B, and their sum with C must equal 180° as a built-in verification check.

SAS — Two Sides and Included Angle Known

Use the Law of Cosines to find the unknown third side, then apply the Law of Sines for the remaining angles. With a = 8, b = 6, C = 40°: c² = 64 + 36 − 2(8)(6)cos(40°) ≈ 100 − 73.47 ≈ 26.53, giving c ≈ 5.15. From c, the Law of Sines provides angles A and B straightforwardly.

ASA — Two Angles and the Included Side Known

Compute C = 180° − A − B immediately, then apply the Law of Sines for both missing sides. Example: A = 50°, B = 60°, c = 12. C = 70°. Side a = 12 × sin(50°) / sin(70°) ≈ 9.79; side b = 12 × sin(60°) / sin(70°) ≈ 11.07. Both computations require only a single Law of Sines ratio pass.

AAS — Two Angles and a Non-Included Side Known

Find C = 180° − A − B, then use the Law of Sines with the given side. Example: A = 35°, B = 75°, a = 10. C = 70°. Side b = 10 × sin(75°) / sin(35°) ≈ 16.85; side c = 10 × sin(70°) / sin(35°) ≈ 16.38. All three sides and all three angles are now fully determined.

Real-World Applications

  • Land Surveying: Surveyors construct oblique triangles across rivers or cliffs to compute distances that cannot be directly measured, using two baseline points and two observed angles.
  • Aviation: Pilots resolve the oblique triangle formed by the airspeed vector, wind vector, and resultant ground-track vector to determine exact heading and ground speed.
  • Structural Engineering: Non-right-angle trusses in roofs and bridges require the Law of Cosines to accurately resolve internal member forces under load.
  • Astronomy: Stellar parallax measurements use an oblique triangle whose base spans the Earth's orbital diameter (approximately 299 million km) to calculate distances to nearby stars in parsecs.

Methodology and Sources

The formulas and case-selection logic implemented in this calculator follow derivations documented in the Clark University Trigonometry Reference and the Colorado State University Trigonometry Placement Guide. Both sources confirm that the Law of Sines, the Law of Cosines, and the trigonometric area formula collectively constitute the complete toolkit for solving any oblique triangle. Weatherford College's Plane Trigonometry (MATH 1316) and Portland Community College's MTH 112 course outcomes further validate this three-formula framework as the standard across university-level trigonometry instruction worldwide.

Reference

Frequently asked questions

What is an oblique triangle and how does it differ from a right triangle?
An oblique triangle is any triangle in which none of the three interior angles measures exactly 90°. Right triangles contain one 90° angle, enabling simpler sine, cosine, and tangent ratios tied to a fixed hypotenuse. Oblique triangles — whether acute (all angles below 90°) or obtuse (one angle above 90°) — require the Law of Sines or the Law of Cosines, both of which apply to any triangle regardless of angle size or side length.
When should the Law of Cosines be used instead of the Law of Sines?
Use the Law of Cosines when the known data consists of all three sides (SSS) or two sides with their included angle (SAS). Use the Law of Sines when two angles are already known alongside any side (ASA or AAS). Starting with the Law of Cosines for SSS and SAS avoids the ambiguous-case problem that can arise when applying the Law of Sines to side-heavy inputs before both angles are established.
How do you solve an oblique triangle when all three sides are known (SSS)?
Apply the Law of Cosines to find the largest angle first: cos(C) = (a² + b² − c²) / (2ab). For sides a = 5, b = 7, c = 9, this gives cos(C) = (25 + 49 − 81) / 70 ≈ −0.1, so C ≈ 95.74°. Next, use the Law of Sines to find one remaining angle, then subtract both from 180° to find the third. Always verify that all three angles sum exactly to 180° as a correctness check.
Can an oblique triangle have two valid solutions?
Yes. The ambiguous SSA case — two sides and a non-included angle — can yield zero, one, or two valid triangles. When the side opposite the known angle is shorter than the altitude h = b · sin(A), no triangle is possible. When it equals h exactly, one right triangle forms. When it falls between h and side b, two distinct oblique triangles both satisfy the given measurements, and the oblique triangle calculator reports both complete solutions.
How is the area of an oblique triangle calculated without knowing its height?
The trigonometric area formula Area = ½ · a · b · sin(C) requires only two sides and the included angle between them — no altitude measurement needed. For a triangle with sides a = 7 m and b = 10 m and included angle C = 45°, the area equals ½ × 7 × 10 × sin(45°) ≈ 24.75 m². The formula is derived by expressing the triangle's height as b · sin(C) and substituting into the standard base-times-height formula.
What real-world problems use oblique triangle calculations?
Oblique triangle formulas appear across multiple technical fields. Land surveyors compute inaccessible boundary lengths by establishing a measured baseline and sighting two angles to a distant target point. Aviation navigators solve the oblique triangle formed by airspeed, wind, and ground-track vectors to find exact heading corrections. Structural engineers determine member forces in angled roof trusses. Astronomers measure stellar distances using the parallax triangle whose base is Earth's orbital diameter of approximately 299 million km.