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TrianglesGeometryHigh schoolabout 2:12

Two sides of a triangle are 6 and 9 with a 120° angle between them. Find the third side: animated lesson

A narrated, step-by-step triangles video lesson with captions. Final answer: c = 3√19 ≈ 13.07669683; A ≈ 23.413224°, B ≈ 36.586776°.

Video with a synthetic English voice and captions. Pick another language for subtitles in that language, or open the interactive version.

Video narrated by a synthetic voice (Piper TTS, en_US-ljspeech-high voice (LJ Speech dataset, public domain)); the steps come from our maths engine and the answer was checked independently. Download MP4 (0.8 MB) · Captions (WebVTT)

Steps on screen

  1. 1. The problem
    Worked example
    Two sides of a triangle are 6 and 9 with a 120° angle between them. Find the third side.
  2. 2. Steps, part 1
    1. c2=62+92−2(6)(9)cos⁡120∘=171c^2 = 6^2 + 9^2 - 2(6)(9)\cos 120^\circ = 171
      Substitute into c² = a² + b² − 2ab·cos C
    2. c=171=319≈13.07669683c = \sqrt{171} = 3\sqrt{19} \approx 13.07669683
      Take the square root
    3. cos⁡A=b2+c2−a22bc≈0.91766294⇒A≈23.413224∘\cos A = \frac{b^2 + c^2 - a^2}{2bc} \approx 0.91766294 \Rightarrow A \approx 23.413224^\circ
      Find angle A with the law of cosines rearranged
    4. B=180∘−A−C≈36.586776∘B = 180^\circ - A - C \approx 36.586776^\circ
      Angles of a triangle add to 180°
  3. 3. Steps, part 2
    1. Area=12absin⁡C≈23.382686\text{Area} = \tfrac12ab\sin C \approx 23.382686
      Area from two sides and the included angle
  4. 4. The answer
    c = 3√19 ≈ 13.07669683; A ≈ 23.413224°, B ≈ 36.586776°\text{c = 3√19 ≈ 13.07669683; A ≈ 23.413224°, B ≈ 36.586776°}

    c = 3√19 ≈ 13.07669683; A ≈ 23.413224°, B ≈ 36.586776°

  5. 5. Check
    • A + B + C = 180°.
    • Heron's formula gives the same area: 23.382686.
    • Law of sines ratios: a/sin A ≈ 15.099669, b/sin B ≈ 15.099669, c/sin C ≈ 15.099669.

Transcript

The problem. Here is the problem: Two sides of a triangle are 6 and 9 with a 120° angle between them. Find the third side. SAS: find the third side and the remaining angles.

Steps, part 1. Substitute into c squared equals a squared plus b squared minus 2ab times cos C. Take the square root. Find angle A with the law of cosines rearranged. Angles of a triangle add to 180°.

Steps, part 2. Area from two sides and the included angle.

The answer. So the answer is c equals 3the square root of 19 approximately 13.07669683; A approximately 23.413224°, B approximately 36.586776°. The triangle has sides a equals 6, b equals 9, c approximately 13.076697 and angles A approximately 23.413224°, B approximately 36.586776°, C approximately 120°. It is an obtuse scalene triangle with area approximately 23.382686 and perimeter approximately 28.076697.

Check. A plus B plus C equals 180°. Heron prime s formula gives the same area: 23.382686. Law of sines ratios: a/sin A approximately 15.099669, b/sin B approximately 15.099669, c/sin C approximately 15.099669.

About triangles

Solve triangles with the law of sines and the law of cosines, and find their area.

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