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TrianglesGeometryHigh schoolabout 2:00

Find all three angles of a triangle with sides 4, 4 and 7: animated lesson

A narrated, step-by-step triangles video lesson with captions. Final answer: A ≈ 28.955024°, B ≈ 28.955024°, C ≈ 122.08995°.

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 4, 4 and 7.
  2. 2. Steps, part 1
    1. Check the triangle inequality
    2. cos⁡C=a2+b2−c22ab=42+42−722(4)(4)≈−0.53125⇒C≈122.08995∘\cos C = \frac{a^2 + b^2 - c^2}{2ab} = \frac{4^2 + 4^2 - 7^2}{2(4)(4)} \approx -0.53125 \Rightarrow C \approx 122.08995^\circ
      Angle C (opposite side c)
    3. cos⁡A=b2+c2−a22bc=42+72−422(4)(7)≈0.875⇒A≈28.955024∘\cos A = \frac{b^2 + c^2 - a^2}{2bc} = \frac{4^2 + 7^2 - 4^2}{2(4)(7)} \approx 0.875 \Rightarrow A \approx 28.955024^\circ
      Angle A (opposite side a)
    4. B=180∘−A−C≈28.955024∘B = 180^\circ - A - C \approx 28.955024^\circ
      Third angle from the angle sum
  3. 3. Steps, part 2
    1. Area=12absin⁡C≈6.7777209\text{Area} = \tfrac12ab\sin C \approx 6.7777209
      Area from two sides and the included angle
  4. 4. The answer
    A ≈ 28.955024°, B ≈ 28.955024°, C ≈ 122.08995°\text{A ≈ 28.955024°, B ≈ 28.955024°, C ≈ 122.08995°}

    A ≈ 28.955024°, B ≈ 28.955024°, C ≈ 122.08995°

  5. 5. Check
    • A + B + C = 180°.
    • Heron's formula gives the same area: 6.7777209.
    • Law of sines ratios: a/sin A ≈ 8.2623645, b/sin B ≈ 8.2623645, c/sin C ≈ 8.2623645.

Transcript

The problem. Here is the problem: Find all three angles of a triangle with sides 4, 4 and 7. SSS: find all three angles.

Steps, part 1. Check the triangle inequality. 4 plus 4 equals 8 greater than 7, 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 28.955024°, B approximately 28.955024°, C approximately 122.08995°. The triangle has sides a equals 4, b equals 4, c equals 7 and angles A approximately 28.955024°, B approximately 28.955024°, C approximately 122.08995°. It is an obtuse isosceles triangle with area approximately 6.7777209 and perimeter approximately 15.

Check. A plus B plus C equals 180°. Heron prime s formula gives the same area: 6.7777209. Law of sines ratios: a/sin A approximately 8.2623645, b/sin B approximately 8.2623645, c/sin C approximately 8.2623645.

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 4, 4 and 7