CK-12-Pre-Calculus Concepts

4.7. Area of a Triangle http://www.ck12.org

Using the sine function, you can isolatehfor height:

sinC=ha

asinC=h

Substituting into the area formula:

Area=^12 b·h

Area=^12 b·a·sinC

Area=^12 ·a·b·sinC

Example A
Given∆ABCwithA= 22 ◦,b= 6 ,c=7. What is the area?
Solution: The letters don’t have to match exactly because the triangle or the formula can just be relabeled. The
important part is that neither given side corresponds to the given angle.

Area=^12 bcsinA

Area=^12 · 6 · 7 ·sin 22◦≈ 7. 86 ...units^2

Example B
Given∆XY Zhas area 28 square inches, what is the angle included between side length 8 and 9?
Solution:

Area=^12 ·a·b·sinC

28 =^12 · 8 · 9 ·sinC

sinC=^288 ·· 92

C=sin−^1

( 28 · 2

8 · 9

)

≈ 51. 05 ...◦

Example C
Given triangleABCwithA= 12 ◦,b=4 andArea= 1. 7 un^2 , what is the length of sidec?
Solution:

Area=^12 ·c·b·sinA

1. 7 =^12 ·c· 4 ·sin 12◦

c= 4 ·^1 sin 12.^7 ·^2 ◦≈ 4. 08 ...