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TrianglesGeometryHigh schoolabout 2:06

Two sides of a triangle are 5 and 7 with a 60° angle between them. Find the third side: animated lesson

A narrated, step-by-step triangles video lesson with captions. Final answer: c = √39 ≈ 6.244997998; A ≈ 43.897886°, B ≈ 76.102114°.

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 5 and 7 with a 60° angle between them. Find the third side.
  2. 2. Steps, part 1
    1. c2=52+72−2(5)(7)cos⁡60∘=39c^2 = 5^2 + 7^2 - 2(5)(7)\cos 60^\circ = 39
      Substitute into c² = a² + b² − 2ab·cos C
    2. c=39=39≈6.244997998c = \sqrt{39} = \sqrt{39} \approx 6.244997998
      Take the square root
    3. cos⁡A=b2+c2−a22bc≈0.72057669⇒A≈43.897886∘\cos A = \frac{b^2 + c^2 - a^2}{2bc} \approx 0.72057669 \Rightarrow A \approx 43.897886^\circ
      Find angle A with the law of cosines rearranged
    4. B=180∘−A−C≈76.102114∘B = 180^\circ - A - C \approx 76.102114^\circ
      Angles of a triangle add to 180°
  3. 3. Steps, part 2
    1. Area=12absin⁡C≈15.155445\text{Area} = \tfrac12ab\sin C \approx 15.155445
      Area from two sides and the included angle
  4. 4. The answer
    c = √39 ≈ 6.244997998; A ≈ 43.897886°, B ≈ 76.102114°\text{c = √39 ≈ 6.244997998; A ≈ 43.897886°, B ≈ 76.102114°}

    c = √39 ≈ 6.244997998; A ≈ 43.897886°, B ≈ 76.102114°

  5. 5. Check
    • A + B + C = 180°.
    • Heron's formula gives the same area: 15.155445.
    • Law of sines ratios: a/sin A ≈ 7.2111026, b/sin B ≈ 7.2111026, c/sin C ≈ 7.2111026.

Transcript

The problem. Here is the problem: Two sides of a triangle are 5 and 7 with a 60° 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 equals the square root of 39 approximately 6.244997998; A approximately 43.897886°, B approximately 76.102114°. The triangle has sides a equals 5, b equals 7, c approximately 6.244998 and angles A approximately 43.897886°, B approximately 76.102114°, C approximately 60°. It is an acute scalene triangle with area approximately 15.155445 and perimeter approximately 18.244998.

Check. A plus B plus C equals 180°. Heron prime s formula gives the same area: 15.155445. Law of sines ratios: a/sin A approximately 7.2111026, b/sin B approximately 7.2111026, c/sin C approximately 7.2111026.

About triangles

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

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