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TrianglesGeometryHigh schoolabout 2:06

Two sides of a triangle are 2.5 and 4 with a 35.5° angle between them. Find the third side: animated lesson

A narrated, step-by-step triangles video lesson with captions. Final answer: c ≈ 2.4428855; A ≈ 36.461304°, B ≈ 108.0387°.

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 (0.8 MB) · Captions (WebVTT)

Steps on screen

  1. 1. The problem
    Worked example
    Two sides of a triangle are 2.5 and 4 with a 35.5° angle between them. Find the third side.
  2. 2. Steps, part 1
    1. c2=2.52+42−2(2.5)(4)cos⁡35.5∘≈5.9676896c^2 = 2.5^2 + 4^2 - 2(2.5)(4)\cos 35.5^\circ \approx 5.9676896
      Substitute into c² = a² + b² − 2ab·cos C
    2. c≈2.4428855c \approx 2.4428855
      Take the square root
    3. cos⁡A=b2+c2−a22bc≈0.8042584⇒A≈36.461304∘\cos A = \frac{b^2 + c^2 - a^2}{2bc} \approx 0.8042584 \Rightarrow A \approx 36.461304^\circ
      Find angle A with the law of cosines rearranged
    4. B=180∘−A−C≈108.0387∘B = 180^\circ - A - C \approx 108.0387^\circ
      Angles of a triangle add to 180°
  3. 3. Steps, part 2
    1. Area=12absin⁡C≈2.9035148\text{Area} = \tfrac12ab\sin C \approx 2.9035148
      Area from two sides and the included angle
  4. 4. The answer
    c ≈ 2.4428855; A ≈ 36.461304°, B ≈ 108.0387°\text{c ≈ 2.4428855; A ≈ 36.461304°, B ≈ 108.0387°}

    c ≈ 2.4428855; A ≈ 36.461304°, B ≈ 108.0387°

  5. 5. Check
    • A + B + C = 180°.
    • Heron's formula gives the same area: 2.9035148.
    • Law of sines ratios: a/sin A ≈ 4.206773, b/sin B ≈ 4.206773, c/sin C ≈ 4.206773.

Transcript

The problem. Here is the problem: Two sides of a triangle are 2.5 and 4 with a 35.5° angle between them. Find the third side. SAS: find the third side and the remaining angles.

Steps, part 1. Substitute into c squared equals a squared plus b squared minus 2ab times cos C. Take the square root. Find angle A with the law of cosines rearranged. Angles of a triangle add to 180°.

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

The answer. So the answer is c approximately 2.4428855; A approximately 36.461304°, B approximately 108.0387°. The triangle has sides a equals 2.5, b equals 4, c approximately 2.4428855 and angles A approximately 36.461304°, B approximately 108.0387°, C approximately 35.5°. It is an obtuse scalene triangle with area approximately 2.9035148 and perimeter approximately 8.9428855.

Check. A plus B plus C equals 180°. Heron prime s formula gives the same area: 2.9035148. Law of sines ratios: a/sin A approximately 4.206773, b/sin B approximately 4.206773, c/sin C approximately 4.206773.

About triangles

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

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