Draw a triangle with side lengths of 11, 14, and 10.
B
b
Recall the formula for the area of a triangle using sine. Note that since we are given the lengths of all three sides of the picnic area, we can use the Law of Cosines to find any angle measure.
C
c
Find the measure of one of the angles using the Law of Cosines. Then use the formula for the area of a triangle using sine.
A
a
Example Sketch:
B
b
See solution.
C
c
54.6 yd^2
Practice makes perfect
a
We are told that a triangular picnic area is 11 yards by 14 yards by 10 yards. First, we want to sketch a drawing to represent this picnic area. Let's plot a triangle with side lengths of 11, 14 and 10.
Note that since we do not know anything specific about the triangle, there is more than one correct way to draw it.
b
Let's now think about how we can find the area of the triangular picnic area. Start by recalling the formula for the area of the triangle using sine.
Area=1/2 b csin ATo use this formula, we need to know two side lengths of a triangle and the measure of their included angle. Consider our diagram.
We are given all three side lengths of this triangle. Two find the measure of any angle we can use the Law of Cosines. For simplicity we will choose ∠A, because it is the largest angle.
Therefore, to find the area of a triangular picnic area we can find the measure of the largest angle using the Law of Cosines. Then we can use this measure and the side lengths of the two sides between which the angle is included in the formula for the area of a triangle using sine.
c
Now, we will find the area of a triangular picnic area. Start by using the Law of Cosines to find the measure of the angle ∠A. Let's substitute a= 14, b= 11, and c= 10 to isolate cos A.
