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TrianglesGeometryHigh schoolabout 1:36

Solve the triangle with a = 12, b = 9 and included angle C = 110°: animated lesson

A narrated, step-by-step triangles video lesson with captions. Final answer: a = 12, b = 9, c = 17.28804069; A = 40.71227504°, B = 29.28772496°, C = 110°; area = 50.74340152, perimeter = 38.28804069.

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 (0.6 MB) · Captions (WebVTT)

Steps on screen

  1. 1. The problem
    Worked example
    Solve the triangle with a = 12, b = 9 and included angle C = 110°.
  2. 2. Step by step
    1. c2=a2+b2−2abcos⁡C=122+92−2(12)(9)cos⁡110∘⇒c≈17.28804069c^2 = a^2 + b^2 - 2ab\cos C = 12^2 + 9^2 - 2(12)(9)\cos 110^\circ \Rightarrow c \approx 17.28804069
      Side c from the law of cosines
    2. A≈40.71227504∘, B≈29.28772496∘A \approx 40.71227504^\circ,\ B \approx 29.28772496^\circ
      Other angles from the law of cosines
    3. Area=12absin⁡C≈50.74340152, P≈38.28804069\text{Area} = \tfrac12ab\sin C \approx 50.74340152,\ P \approx 38.28804069
      Area and perimeter
  3. 3. The answer
    Area≈50.74340152\text{Area} \approx 50.74340152

    area ≈ 50.74340152

  4. 4. Check
    • Triangle 1: angles sum to 180°; a/sin A = 18.39754863 = c/sin C = 18.39754863.

Transcript

The problem. Here is the problem: Solve the triangle with a equals 12, b equals 9 and included angle C equals 110°. Side a is opposite angle A, b opposite B and c opposite C. This is the SAS case.

Step by step. Side c from the law of cosines. Other angles from the law of cosines. The law of cosines avoids the sine ambiguity for obtuse angles. Area and perimeter.

The answer. So the answer is a equals 12, b equals 9, c equals 17.28804069; A equals 40.71227504°, B equals 29.28772496°, C equals 110°; area equals 50.74340152, perimeter equals 38.28804069. Three measurements including a side normally fix a triangle. Two sides with a non-included angle (SSA) can give zero, one or two triangles.

Check. Triangle 1: angles sum to 180°; a/sin A equals 18.39754863 equals c/sin C equals 18.39754863.

About triangles

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

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