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TrianglesGeometryHigh schoolabout 1:33

Solve the triangle with a = 5, b = 8 and included angle C = 40°: animated lesson

A narrated, step-by-step triangles video lesson with captions. Final answer: a = 5, b = 8, c = 5.264640971; A = 37.62394377°, B = 102.3760562°, C = 40°; area = 12.85575219, perimeter = 18.26464097.

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 = 5, b = 8 and included angle C = 40°.
  2. 2. Step by step
    1. c2=a2+b2−2abcos⁡C=52+82−2(5)(8)cos⁡40∘⇒c≈5.264640971c^2 = a^2 + b^2 - 2ab\cos C = 5^2 + 8^2 - 2(5)(8)\cos 40^\circ \Rightarrow c \approx 5.264640971
      Side c from the law of cosines
    2. A≈37.62394377∘, B≈102.3760562∘A \approx 37.62394377^\circ,\ B \approx 102.3760562^\circ
      Other angles from the law of cosines
    3. Area=12absin⁡C≈12.85575219, P≈18.26464097\text{Area} = \tfrac12ab\sin C \approx 12.85575219,\ P \approx 18.26464097
      Area and perimeter
  3. 3. The answer
    Area≈12.85575219\text{Area} \approx 12.85575219

    area ≈ 12.85575219

  4. 4. Check
    • Triangle 1: angles sum to 180°; a/sin A = 8.190327398 = c/sin C = 8.190327398.

Transcript

The problem. Here is the problem: Solve the triangle with a equals 5, b equals 8 and included angle C equals 40°. 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 5, b equals 8, c equals 5.264640971; A equals 37.62394377°, B equals 102.3760562°, C equals 40°; area equals 12.85575219, perimeter equals 18.26464097. 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 8.190327398 equals c/sin C equals 8.190327398.

About triangles

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

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