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TrianglesGeometryHigh schoolabout 1:30

Find all three angles of a triangle with sides 3, 4 and 5: animated lesson

A narrated, step-by-step triangles video lesson with captions. Final answer: A ≈ 36.869898°, B ≈ 53.130102°, C ≈ 90°.

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
    Find all three angles of a triangle with sides 3, 4 and 5.
  2. 2. Steps, part 1
    1. Check the triangle inequality
    2. cos⁡C=a2+b2−c22ab=32+42−522(3)(4)≈0⇒C≈90∘\cos C = \frac{a^2 + b^2 - c^2}{2ab} = \frac{3^2 + 4^2 - 5^2}{2(3)(4)} \approx 0 \Rightarrow C \approx 90^\circ
      Angle C (opposite side c)
    3. cos⁡A=b2+c2−a22bc=42+52−322(4)(5)≈0.8⇒A≈36.869898∘\cos A = \frac{b^2 + c^2 - a^2}{2bc} = \frac{4^2 + 5^2 - 3^2}{2(4)(5)} \approx 0.8 \Rightarrow A \approx 36.869898^\circ
      Angle A (opposite side a)
    4. B=180∘−A−C≈53.130102∘B = 180^\circ - A - C \approx 53.130102^\circ
      Third angle from the angle sum
  3. 3. Steps, part 2
    1. Area=12absin⁡C≈6\text{Area} = \tfrac12ab\sin C \approx 6
      Area from two sides and the included angle
  4. 4. The answer
    A ≈ 36.869898°, B ≈ 53.130102°, C ≈ 90°\text{A ≈ 36.869898°, B ≈ 53.130102°, C ≈ 90°}

    A ≈ 36.869898°, B ≈ 53.130102°, C ≈ 90°

  5. 5. Check
    • A + B + C = 180°.
    • Heron's formula gives the same area: 6.
    • Law of sines ratios: a/sin A ≈ 5, b/sin B ≈ 5, c/sin C ≈ 5.

Transcript

The problem. Here is the problem: Find all three angles of a triangle with sides 3, 4 and 5. SSS: find all three angles.

Steps, part 1. Check the triangle inequality. 3 plus 4 equals 7 greater than 5, 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 36.869898°, B approximately 53.130102°, C approximately 90°. The triangle has sides a equals 3, b equals 4, c equals 5 and angles A approximately 36.869898°, B approximately 53.130102°, C approximately 90°. It is a right scalene triangle with area approximately 6 and perimeter approximately 12.

Check. A plus B plus C equals 180°. Heron prime s formula gives the same area: 6. Law of sines ratios: a/sin A approximately 5, b/sin B approximately 5, c/sin C approximately 5.

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 3, 4 and 5