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TrianglesGeometryHigh schoolabout 1:32

Solve the triangle with a = 7, b = 10 and included angle C = 60°: animated lesson

A narrated, step-by-step triangles video lesson with captions. Final answer: a = 7, b = 10, c = 8.888194417; A = 43.00391194°, B = 76.99608806°, C = 60°; area = 30.31088913, perimeter = 25.88819442.

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
    Solve the triangle with a = 7, b = 10 and included angle C = 60°.
  2. 2. Step by step
    1. c2=a2+b2−2abcos⁡C=72+102−2(7)(10)cos⁡60∘⇒c≈8.888194417c^2 = a^2 + b^2 - 2ab\cos C = 7^2 + 10^2 - 2(7)(10)\cos 60^\circ \Rightarrow c \approx 8.888194417
      Side c from the law of cosines
    2. A≈43.00391194∘, B≈76.99608806∘A \approx 43.00391194^\circ,\ B \approx 76.99608806^\circ
      Other angles from the law of cosines
    3. Area=12absin⁡C≈30.31088913, P≈25.88819442\text{Area} = \tfrac12ab\sin C \approx 30.31088913,\ P \approx 25.88819442
      Area and perimeter
  3. 3. The answer
    Area≈30.31088913\text{Area} \approx 30.31088913

    area ≈ 30.31088913

  4. 4. Check
    • Triangle 1: angles sum to 180°; a/sin A = 10.26320288 = c/sin C = 10.26320288.

Transcript

The problem. Here is the problem: Solve the triangle with a equals 7, b equals 10 and included angle C equals 60°. Side a is opposite angle A, b opposite B and c opposite C. This is the SAS case.

Step by step. Side c from the law of cosines. Other angles from the law of cosines. The law of cosines avoids the sine ambiguity for obtuse angles. Area and perimeter.

The answer. So the answer is a equals 7, b equals 10, c equals 8.888194417; A equals 43.00391194°, B equals 76.99608806°, C equals 60°; area equals 30.31088913, perimeter equals 25.88819442. Three measurements including a side normally fix a triangle. Two sides with a non-included angle (SSA) can give zero, one or two triangles.

Check. Triangle 1: angles sum to 180°; a/sin A equals 10.26320288 equals c/sin C equals 10.26320288.

About triangles

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

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