SAS is a term used for triangles that means side angle side. It’s a rule used for finding the last measurement of a triangle. In other words, if two sides are known and the third needs to be found the SAS rule can be used.
There are three steps in the SAS solution for triangles. Each involves a rule or law with an equation to solve.
If two sides are known, the angle that connects them is known as well. That gives three pieces of information to help find the last side. This information can be used in the Law of Cosigns, which looks like this, c2 = a2 + b2 – 2ab cos(C).
The information from the two known sides can be added to the equation to find the angles of the two remaining corners.
With the information available the equation would be set up as, c2 = 82 + 112 – 2-8 x 11 x cos(37). The equation can be reduced by following the rule of operations to leave c2 = 64 + 121 – 176 x 0.798. Solving the equation could prove that c = 6.67. If side a is 8 inches and side b is 11 inches, side c is 6.67 inches.
The law of sines can use the information from the equation to find the unknown angle. It’s a bit lengthy, but the operations are simple. The equation can also be flipped around to find either the side opposite of a known angle or an angle opposite of a known side.
Information that’s already know can be added. The known angle is 63o , one of the known sides is 5.5 inches, and the other known side is 4.7 inches.
The equation would look like this, sin A / a = sin B / 4.7 = sin 63 / 5.5 and can then be reduced by eliminating A / a since it has no value. B / 4.7 = sin 63 / 5.5 can be reduced further by multiplying both sides and solving for sin 63 / 5.5, leaving the answer for B as 0.7614. The last step is to inverse sin, the formula would look like this sin-1 ( 0.7614 ) = 49.6o, making the second angle known.
The last step is the easiest of the SAS functions. This is the rule that all interior angles of a triangle add up to 180o. If the first known angle is 63o and the second known angle is 49.6o, the third angle can be found by subtracting the to known angles from 180o. The equation would look like this, A + B + C + 180o. Since two angles are known it would look like this 112.6 + C = 180. Inversing the equation and solving it would leave C = 67.4o.
Since all three angles are known the triangle can be drawn and the last side can be measured. The information found by solving an SAS triangle can be used to calculate area and other equations that might be needed for additional operations.