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TrianglesGeometryHigh schoolabout 1:53

Find all three angles of a triangle with sides 7, 8 and 13: animated lesson

A narrated, step-by-step triangles video lesson with captions. Final answer: A ≈ 27.795772°, B ≈ 32.204228°, C ≈ 120°.

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 7, 8 and 13.
  2. 2. Steps, part 1
    1. Check the triangle inequality
    2. cos⁡C=a2+b2−c22ab=72+82−1322(7)(8)≈−0.5⇒C≈120∘\cos C = \frac{a^2 + b^2 - c^2}{2ab} = \frac{7^2 + 8^2 - 13^2}{2(7)(8)} \approx -0.5 \Rightarrow C \approx 120^\circ
      Angle C (opposite side c)
    3. cos⁡A=b2+c2−a22bc=82+132−722(8)(13)≈0.88461538⇒A≈27.795772∘\cos A = \frac{b^2 + c^2 - a^2}{2bc} = \frac{8^2 + 13^2 - 7^2}{2(8)(13)} \approx 0.88461538 \Rightarrow A \approx 27.795772^\circ
      Angle A (opposite side a)
    4. B=180∘−A−C≈32.204228∘B = 180^\circ - A - C \approx 32.204228^\circ
      Third angle from the angle sum
  3. 3. Steps, part 2
    1. Area=12absin⁡C≈24.248711\text{Area} = \tfrac12ab\sin C \approx 24.248711
      Area from two sides and the included angle
  4. 4. The answer
    A ≈ 27.795772°, B ≈ 32.204228°, C ≈ 120°\text{A ≈ 27.795772°, B ≈ 32.204228°, C ≈ 120°}

    A ≈ 27.795772°, B ≈ 32.204228°, C ≈ 120°

  5. 5. Check
    • A + B + C = 180°.
    • Heron's formula gives the same area: 24.248711.
    • Law of sines ratios: a/sin A ≈ 15.011107, b/sin B ≈ 15.011107, c/sin C ≈ 15.011107.

Transcript

The problem. Here is the problem: Find all three angles of a triangle with sides 7, 8 and 13. SSS: find all three angles.

Steps, part 1. Check the triangle inequality. 7 plus 8 equals 15 greater than 13, 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 27.795772°, B approximately 32.204228°, C approximately 120°. The triangle has sides a equals 7, b equals 8, c equals 13 and angles A approximately 27.795772°, B approximately 32.204228°, C approximately 120°. It is an obtuse scalene triangle with area approximately 24.248711 and perimeter approximately 28.

Check. A plus B plus C equals 180°. Heron prime s formula gives the same area: 24.248711. Law of sines ratios: a/sin A approximately 15.011107, b/sin B approximately 15.011107, c/sin C approximately 15.011107.

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 7, 8 and 13