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Geometry · Triangles

Triangle 3 5 7

3,\ 5,\ 7

Step by step

  1. a = 3,\ b = 5,\ c = 7

    Three sides (SSS). Check the triangle inequality: each side is less than the sum of the other two. ✓

  2. P = a + b + c = 15

    Perimeter.

  3. s = \tfrac{P}{2} = \frac{15}{2},\quad A = \sqrt{s(s-a)(s-b)(s-c)} = \frac{15 \sqrt{3}}{4} \approx 6.4952

    Heron's formula for the area.

  4. \cos A = \frac{b^2 + c^2 - a^2}{2bc} = \frac{13}{14} \Rightarrow A \approx 21.787^\circ

    Law of cosines for angle A (opposite side a).

  5. \cos B = \frac{a^2 + c^2 - b^2}{2ac} = \frac{11}{14} \Rightarrow B \approx 38.213^\circ

    Law of cosines for angle B (opposite side b).

  6. \cos C = \frac{a^2 + b^2 - c^2}{2ab} = - \frac{1}{2} \Rightarrow C \approx 120.00^\circ

    Law of cosines for angle C (opposite side c).

  7. A + B + C = 180.0^\circ

    The angles add to 180° — a obtuse, scalene triangle.

Reveal the answer
A = \frac{15 \sqrt{3}}{4} \approx 6.4952,\quad P = 15,\quad \angle \approx 21.8^\circ, 38.2^\circ, 120.0^\circ