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TrianglesGeometryHigh schoolabout 2:05

Two sides of a triangle are 3 and 8 with a 45° angle between them. Find the third side: animated lesson

A narrated, step-by-step triangles video lesson with captions. Final answer: c ≈ 6.24971; A ≈ 19.841968°, B ≈ 115.15803°.

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 3 and 8 with a 45° angle between them. Find the third side.
  2. 2. Steps, part 1
    1. c2=32+82−2(3)(8)cos⁡45∘≈39.058875c^2 = 3^2 + 8^2 - 2(3)(8)\cos 45^\circ \approx 39.058875
      Substitute into c² = a² + b² − 2ab·cos C
    2. c≈6.24971c \approx 6.24971
      Take the square root
    3. cos⁡A=b2+c2−a22bc≈0.9406324⇒A≈19.841968∘\cos A = \frac{b^2 + c^2 - a^2}{2bc} \approx 0.9406324 \Rightarrow A \approx 19.841968^\circ
      Find angle A with the law of cosines rearranged
    4. B=180∘−A−C≈115.15803∘B = 180^\circ - A - C \approx 115.15803^\circ
      Angles of a triangle add to 180°
  3. 3. Steps, part 2
    1. Area=12absin⁡C≈8.4852814\text{Area} = \tfrac12ab\sin C \approx 8.4852814
      Area from two sides and the included angle
  4. 4. The answer
    c ≈ 6.24971; A ≈ 19.841968°, B ≈ 115.15803°\text{c ≈ 6.24971; A ≈ 19.841968°, B ≈ 115.15803°}

    c ≈ 6.24971; A ≈ 19.841968°, B ≈ 115.15803°

  5. 5. Check
    • A + B + C = 180°.
    • Heron's formula gives the same area: 8.4852814.
    • Law of sines ratios: a/sin A ≈ 8.8384246, b/sin B ≈ 8.8384246, c/sin C ≈ 8.8384246.

Transcript

The problem. Here is the problem: Two sides of a triangle are 3 and 8 with a 45° 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 6.24971; A approximately 19.841968°, B approximately 115.15803°. The triangle has sides a equals 3, b equals 8, c approximately 6.24971 and angles A approximately 19.841968°, B approximately 115.15803°, C approximately 45°. It is an obtuse scalene triangle with area approximately 8.4852814 and perimeter approximately 17.24971.

Check. A plus B plus C equals 180°. Heron prime s formula gives the same area: 8.4852814. Law of sines ratios: a/sin A approximately 8.8384246, b/sin B approximately 8.8384246, c/sin C approximately 8.8384246.

About triangles

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

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