Area of Triangles

Area of Triangles

Open Ended

What do you know about finding the area of a triangle?

Area of Right Triangle

•  $\frac{1}{2}$  base x height 

•  $\frac{1}{2}ba$  

Area of Triangle SAS

• Area= $\frac{1}{2}$   ( 2 adjacent sides and the sin of the included angle)

• For the give triangle,  $A=\frac{1}{2}ab\sin C$  

• Could also be  $A=\frac{1}{2}bcSinA$  ,  $A=\frac{1}{2}ac\sin B$  

Example

• Find the area of triangle ABC with $\angle A=40\degree,\ b=4cm,\ and\ c=8cm$   

•  $A=\frac{1}{2}bc\sin A$  

•  $A=\frac{1}{2}\left(8\right)\left(4\right)\sin40$  

•  $A=10.3cm^2$  

Multiple Choice

Find the area of a triangle with $\angle B=20\degree,\ a=10\ cm,\ c=4cm$  

•  $6.8cm^2$  

•  $12cm^2$  

•  $25cm^2$  

•  $15cm^2$  

Example 2

• The area of triangle XYZ is $36cm^2$ . If side x=12 and y=8 find the measurements of the included angles Z.

•  $A=\frac{1}{2}xy\sin Z$  

•  $36=\frac{1}{2}\left(12\right)\left(8\right)\sin Z$  

•  $36=48\sin Z$  

•  $\frac{3}{4}=\sin Z$  (since positive sin the angles  can be found in the first and second quadrant

• Z= $48.6\degree\ or\ 180-48.6=131.4\degree$  

Multiple Choice

The area of triangle DEF is $15m^2$ . If e=9m and f=4m, what are the possible angle measurements of angle D?

•  $45\degree,134.3\degree$  

•  $56.4\degree,\ 123.6\degree$  

•  $53.7\degree,\ 105\degree$  

•  $67.2\degree,\ 184\degree$  

Area of Triangle SSS

• Finding the area of a triangle if given 3 side lengths

• Use Heron's formula $A=\sqrt{s\left(s-a\right)\left(s-b\right)\left(s-c\right)}$   

• Where s= $\frac{a+b+c}{2}$  (semi perimeter)

Example

• Find the area of triangle XYZ if x=2, y=7, and z=8

•  $s=\frac{x+y+z}{2}$  First solve for s.  

•  $s=\frac{2+7+8}{2}$  so s= 8.5

• Substitute values into formula and solve 

•  $A=\sqrt{s\left(s-x\right)\left(s-y\right)\left(s-z\right)}$  

•  $A=\sqrt{8.5\left(8.5-2\right)\left(8.5-7\right)\left(8.5-8\right)}$  

•  $A=\sqrt{41.4375},\ A=6.44$  units squared

Multiple Choice

Using Heron's Formula, find the Area.

• 56

• 28

• 24.2

• 104.4

Multiple Choice

Which of the following is Heron's Formula?

• $A=s\left(s-a\right)\left(s-b\right)\left(s-c\right)$

• $s=\frac{a+b+c}{2}$

• $A=\sqrt{s\left(s-a\right)\left(s-b\right)\left(s-c\right)}$

Multiple Choice

Which of the following is a correct version of the area of triangle (SAS)?

• $A=2bh$

• $A=\frac{1}{2}\sin A$

• $A=\frac{1}{2}bc\sin A$