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Tutorial on Triangles & How to Figure the Unknown Side

There are six different general classifications of triangles: right, equilateral, isosceles, scalene, acute and obtuse. A right triangle has a 90 degree angle and is the most commonly used triangle in math and the sciences. Equilateral triangles have three equal sides and angles. Isosceles triangles have two equal sides and angles. Scalene triangles have no equal sides or angles. Acute triangles have three acute angles, meaning each angle is less than 90 degrees in measure. An obtuse triangle has one obtuse angle, meaning it measures to greater than 90 degrees. All triangles have an angular sum of 180 degrees and can be solved for an unknown side.

Instructions

Right Triangles
- 1
  Draw the triangle and label the two known sides. Remember, the hypotenuse is the longest leg, the base leg runs along the bottom of the triangle and the third leg connects the base to the hypotenuse.
- 2
  Substitute the known side lengths of the triangle into the Pythagorean Theorem: a^2 + b^2 = c^2, where c is the hypotenuse. For example, if you know the length of the base leg equals 5 and the length of the third leg equals 8 then the Pythagorean Theorem equation becomes (5)^2 + (8)^2 = c^2.
- 3
  Solve the equation for the unknown side. For example, if the Pythagorean Theorem equation for a triangle is (5)^2 + (8)^2 = c^2, solving for c finds: (5)^2 + (8)^2 = c^2 ---> 25 + 64 = c^2 ---> 89 = c^2 ---> sqrt(c) = sqrt(89) ---> c = 9.43. This is the length of the unknown leg.
Other Regular Triangles
- 4
  Identify the triangle as isoceles by noting that the triangle has two equal sides.
- 5
  Note that the unknown side length will be the same as the other, equal side length.
- 6
  Identify a triangle as an equilateral by noting that the triangle has three sides of equal length.
- 7
  Note that the unknown side length equals the length of the other sides.
Irregular Triangles
- 8
  Substitute the known side lengths into the law of cosines equation: a = sqrt(b^2 + c^2 - (2)(b)(c) *cos(A), where "a" is the unknown side, "b" and "c" are the known sides and "A" is the angle opposite the unknown side.
- 9
  Solve the law of cosines equation for the unknown side length. For example, if known side lengths are 5 and 9, and the angle opposite the unknown side is 47 degrees, the law of cosines becomes: a = sqrt(5^2 + 9^2 - (2)(5)(9) * cos(47)) = sqrt(25 + 81 - 90 * cos(47)) = sqrt(106 - 61.38) = sqrt(44.62) = 6.68.
- 10
  Confirm the answer by substituting your answer into the law of cosines equation and solve for "A." The law of cosines becomes: A = arccos((b^2 + c^2 - a^2) / (2)(b)(c)), when rearranged to solve for "A."
- 11
  Solve the law of cosines equation for "A." For example, for a scalene triangle with side lengths a = 3.3, b = 5 and c = 9, the equation becomes: A = arccos((5^2 + 9^2 - 6.68^2) / (2)(5)(9)) = arccos((25 + 81 - 44.6) / 90) = arccos(61.4 / 90) = arccos(0.682) = 47 degrees.

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