Dayaan M. answered • 08/26/26
Earned A’s Twice in Precalculus | 5 Years of Tutoring Experience
In order to pick the right area formula, look at what you were given. You have two sides and the angle sitting BETWEEN them, since side a is opposite angle A, side b is opposite angle B, and angle C is the one tucked between sides a and b. That specific arrangement, two sides and the included angle, is exactly what the SAS area formula is for:
Area = (1/2)ab sin(C)
Notice we cannot use the usual one half base times height here, because we were never given a height. That is the whole reason this formula exists. The sin(C) piece is quietly doing the work of finding the height for us.
Before plugging in, we need to deal with the angle notation. That 95 degrees 52 minutes is not 95.52 degrees. A minute is one sixtieth of a degree, the same way a minute is one sixtieth of an hour, so we divide by 60:
52/60 = 0.8667
which makes the angle 95.8667 degrees. Getting this wrong is the most common mistake on this problem, so make sure your calculator is in degree mode and that you converted rather than just moving a decimal point.
Now we substitute:
Area = (1/2)(35)(26) sin(95.8667)
Half of 35 times 26 is 455, and sin(95.8667) is about 0.9947:
Area = 455 x 0.9947 = 452.6
So, our final answer is about 452.6 square units.
One thing worth noticing: the angle is just past 90 degrees, and sine peaks at 90, so sin(C) is very close to 1. That means the area is nearly the maximum it could be for these two side lengths. If you ever get an answer for a problem like this that is far below (1/2)ab, that is a good hint you made an angle mistake somewhere.
