4.7: Area of a Triangle
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From geometry you already know that the area of a triangle is 12⋅b⋅h
What if you are given the sides of a triangle are 5 and 6 and the angle between the sides is π3? You are not directly given the height, but can you still figure out the area of the triangle?
Finding the Area of Triangles
The sine function allows you to find the height of any triangle and substitute that value into the familiar triangle area formula.
Using the sine function, you can isolate h for height:
sinC=haasinC=h
Substituting into the area formula:
Area=12b⋅hArea=12b⋅a⋅sinCArea=12⋅a⋅b⋅sinC
Recall that the variables used to note the sides and corresponding angles are arbitrary, so long as a side and its opposing angle share the same variable (side c has an opposing angle C ). If you were given ΔABC with A=22∘,b=6,c=7 and asked to find the area, you would use the formula:
Area=12bcsinAArea=12⋅6⋅7⋅sin22∘≈7.86 units2
The important part is that neither given side corresponds to the given angle.
Bonus Video: There is another way of finding the area of a triangle, Heron's Formula. This will be discussed in Example 5. Heron's formula is used when three side lengths are given.
Examples
Earlier, you were given the sides of a triangle are 5 and 6 and the angle between the sides is θ=π3 and asked to find the area.
Area =12⋅5⋅6⋅sinπ3≈12.99 units 2
Given ΔXYZ has area 28 square inches, what is the angle included between side length 8 and 9?
Area=12⋅a⋅b⋅sinC28=12⋅8⋅9⋅sinCsinC=28⋅28⋅9C=sin−1(28⋅28⋅9)≈51.06∘
Given triangle ABC with A=12∘,b=4 and Area=1.7 units2, what is the length of side c?
Area=12⋅c⋅b⋅sinA1.7=12⋅c⋅4⋅sin12∘c=1.7⋅24⋅sin12∘≈4.09
The area of a triangle is 3 square units. Two sides of the triangle are 4 units and 5 units. What is the measure of their included angle?
3=12⋅4⋅5⋅sinθ
θ=sin−1(3⋅24⋅5)≈17.46∘
What is the area of ΔXYZ with x=11,y=12,z=13?
Since none of the angles are given, there are two possible solution paths. You could use the Law of Cosines to find one angle.
Area =12⋅12⋅13⋅sin52.02≈61.5 units 2
The angle opposite the side of length 11 is approximately 52.02∘ therefore the area is:
Area =12⋅12⋅13⋅sin52.02≈61.5 units 2
Another way to find the area is through the use of Heron's Formula which is:
Area=√s(s−a)(s−b)(s−c)
Where s is the semiperimeter:
s=a+b+c2
Using Heron's formula to find the area of ΔXYZ returns the same value:
s=a+b+c2
s=11+12+132=362=18
A=√18(18−11)(18−12)(18−13)
A=√18⋅7⋅6⋅5
A=√3780≈61.5 units 2
The area of a triangle is 3 square units. Two sides of the triangle are 4 units and 5 units. What is the measure of their included angle?
3=12⋅4⋅5⋅sinθ
θ=sin−1(3⋅24⋅5)≈17.45…
Review
For 1−11, find the area of each triangle.
1. ΔABC if a=13,b=15, and ∠C=70∘.
2. ΔABC if b=8,c=4, and ∠A=58∘.
3. ΔABC if b=34,c=29, and ∠A=125∘.
4. ΔABC if a=3,b=7, and ∠C=81∘.
5. ΔABC if a=4.8,c=3.7, and ∠B=54∘.
6. ΔABC if a=12,b=5, and ∠C=22∘.
7. ΔABC if a=3,b=10, and ∠C=65∘.
8. ΔABC if a=5,b=9, and ∠C=11∘.
9. ΔABC if a=5,b=7, and c=8.
10. ΔABC if a=7,b=8, and c=14.
11. ΔABC if a=12,b=14, and c=13.
12. The area of a triangle is 12 square units. Two sides of the triangle are 8 units and 4 units. What is the measure of their included angle?
13. The area of a triangle is 23 square units. Two sides of the triangle are 14 units and 5 units. What is the measure of their included angle?
14. Given ΔDEF has area 32 square inches, what is the angle included between side length 9 and 10?
15. Given ΔGHI has area 15 square inches, what is the angle included between side length 7 and 11?