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How to Solve Triangle Equations in Geometry

Triangles are geometric figures with three sides and three internal angles. In an isosceles triangle, two sides of equal length connected to a different sized base. Equilateral triangles have three sides of equal length and all of the angles are also of equal size. A standard right triangle has one 90 degree angle and one longer, slanted side called the hypotenuse. Specialty right triangles have angles of 45, 45 and 90 degrees, or of 30, 60 and 90 degrees.

Instructions

- 1
  Solve the area of a triangle, which can be thought of as half of a polygon, by dividing the area of a polygon by two.
  
  First, find the base length (b) and the height length (h). Multiple the base times the height, as you would for a regular polygon. Then, cut it in two, or divide in half. (b * h) / 2
  
  To solve the area of a triangle with a base of 4 and height of 3:
  
  (4 * 3) / 2. (4*3)=12, then 12 divided by 2 = 6.
  
  The area of the triangle is 6.
- 2
  Calculate an unknown angle using the knowledge that all three angles within a triangle add together to a total of 180 degrees, or angle x + angle y + angle z = 180.
  
  If we have a triangle where angle x is unknown, but we know angle y = 35, and angle z = 50, then we can add those two together to get 85.
  
  Since x + y + z = 180, we can solve x + 85 = 180.
  
  Subtract 85 from both sides, and x = 95 degrees.
- 3
  Find an unknown side of a right triangle, use the Pythagorean Theorem. It states that a^2 + b^2 = c^2 where "c" is the hypotenuse and "a" and "b" are the other two sides.
  
  To solve for an unknown side of a triangle with a hypotenuse (c) of 5 and a side (b) of 4, put those known factors into the equation: a^2 + 4^2 = 5^2.
  
  Simplify: a^2 + 16 = 25.
  
  Subtract 16 from both sides: a^2 = 9. Eliminate the exponent by taking the square root of both sides: a = 3.
- 4
  Calculate an unknown side on a 30-60-90 triangle using the knowledge that the height equals "a," the base equals "a * √3" and the hypotenuse equals "2a."
  
  To solve a triangle with a known height of 4, but the other two sides are unknown, put the known factors into the equation: base = 4 * √3 = 6.93 (rounded) and hypotenuse = 2 * 4 = 8.
  
  Solve a triangle with a hypotenuse of 6 by first solving for "a": 2a = 6 becomes a = 3.
  
  Then solve for the base: 3 * √3 = 5.20 (rounded).
- 5
  Solve a 45-45-90 triangle using the knowledge that the height and base are "a" and the hypotenuse is "a * √2".
  
  Solve a triangle with a height/base of 4: 4 * √2 = 5.66 (rounded).

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