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TrianglesGeometryHigh schoolabout 1:58

Find all three angles of a triangle with sides 5, 6 and 9: animated lesson

A narrated, step-by-step triangles video lesson with captions. Final answer: A ≈ 31.586338°, B ≈ 38.942441°, C ≈ 109.47122°.

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 (0.7 MB) · Captions (WebVTT)

Steps on screen

  1. 1. The problem
    Worked example
    Find all three angles of a triangle with sides 5, 6 and 9.
  2. 2. Steps, part 1
    1. Check the triangle inequality
    2. cos⁡C=a2+b2−c22ab=52+62−922(5)(6)≈−0.33333333⇒C≈109.47122∘\cos C = \frac{a^2 + b^2 - c^2}{2ab} = \frac{5^2 + 6^2 - 9^2}{2(5)(6)} \approx -0.33333333 \Rightarrow C \approx 109.47122^\circ
      Angle C (opposite side c)
    3. cos⁡A=b2+c2−a22bc=62+92−522(6)(9)≈0.85185185⇒A≈31.586338∘\cos A = \frac{b^2 + c^2 - a^2}{2bc} = \frac{6^2 + 9^2 - 5^2}{2(6)(9)} \approx 0.85185185 \Rightarrow A \approx 31.586338^\circ
      Angle A (opposite side a)
    4. B=180∘−A−C≈38.942441∘B = 180^\circ - A - C \approx 38.942441^\circ
      Third angle from the angle sum
  3. 3. Steps, part 2
    1. Area=12absin⁡C≈14.142136\text{Area} = \tfrac12ab\sin C \approx 14.142136
      Area from two sides and the included angle
  4. 4. The answer
    A ≈ 31.586338°, B ≈ 38.942441°, C ≈ 109.47122°\text{A ≈ 31.586338°, B ≈ 38.942441°, C ≈ 109.47122°}

    A ≈ 31.586338°, B ≈ 38.942441°, C ≈ 109.47122°

  5. 5. Check
    • A + B + C = 180°.
    • Heron's formula gives the same area: 14.142136.
    • Law of sines ratios: a/sin A ≈ 9.5459415, b/sin B ≈ 9.5459415, c/sin C ≈ 9.5459415.

Transcript

The problem. Here is the problem: Find all three angles of a triangle with sides 5, 6 and 9. SSS: find all three angles.

Steps, part 1. Check the triangle inequality. 5 plus 6 equals 11 greater than 9, so a triangle exists. Angle C (opposite side c). Angle A (opposite side a). Third angle from the angle sum.

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

The answer. So the answer is A approximately 31.586338°, B approximately 38.942441°, C approximately 109.47122°. The triangle has sides a equals 5, b equals 6, c equals 9 and angles A approximately 31.586338°, B approximately 38.942441°, C approximately 109.47122°. It is an obtuse scalene triangle with area approximately 14.142136 and perimeter approximately 20.

Check. A plus B plus C equals 180°. Heron prime s formula gives the same area: 14.142136. Law of sines ratios: a/sin A approximately 9.5459415, b/sin B approximately 9.5459415, c/sin C approximately 9.5459415.

About triangles

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

More lessons like this

Find all three angles of a triangle with sides 5, 6 and 9