Toby draws a triangle that has two side lengths of 8 inches and 5 inches. He measures the included angle with a protractor and gets
Guidance
Recall the non right triangle for which we derived the law of sine.
We are most familiar with the area formula:
No we can replace
It is important to note that
Example A
Find the area of the triangle.
Solution: We are given two sides and the included angle so let
Example B
Find the area of the triangle.
Solution: In this triangle we do not have two sides and the included angle. We must first find another side length using the Law of Sines. We can find the third angle using the triangle sum:
Put these measures in the triangle:
We now have two sides and the included angle and can use the area formula:
Example C
Given
Solution: Here we are given two sides and the included angle. We can adjust the formula to represent the sides and angle we are given:
Concept Problem Revisit We are given two sides and the included angle so let
Guided Practice
Find the area of each of the triangles below. Round answers to the nearest square unit.
1.
2.
3.
Answers
1. Two sides and the included angle are given so
2. Find side
Using the area formula,
3. Find
Using the area formula,
Explore More
Find the area of each of the triangles below. Round your answers to the nearest square unit.

m∠A=71∘,b=15,c=19 
m∠C=120∘,b=22,a=16 
m∠B=60∘,a=18,c=12 
m∠A=28∘,m∠C=73∘,b=45 
m∠B=56∘,m∠C=81∘,c=33 
m∠A=100∘,m∠B=30∘,a=100  The area of
ΔABC is 66 square units. If two sides of the triangle are 11 and 21 units, what is the measure of the included angle? Is there more than one possible value? Explain.  A triangular garden is bounded on one side by a 20 ft long barn and a second side is bounded by a 25 ft long fence. If the barn and the fence meet at a
50∘ angle, what is the area of the garden if the third side is the length of the segment between the ends of the fence and the barn?  A contractor is constructing a counter top in the shape of an equilateral triangle with side lengths 3 ft. If the countertop material costs $25 per square foot, how much will the countertop cost?