Question

Two similar pentagons have side lengths of 9 inches (OLD) and 15 inches (NEW) what’s is the ratio of their areas?

Answer 1

P1= side 9 inches

P2= side 15 inches

what’s is the ratio of their areas?

the area of a pentagon is:

`\text{area}=\frac{\text{perimeter}\cdot\text{apothem}}{2}`

so first we have to find the perimeters

`\begin{gathered} P1=9\cdot5=45 \\ P2=15\cdot5=75 \end{gathered}`

and now we have to find the apothem

`\begin{gathered} x1 \\ 4.5^2+x1^2=9^2 \\ x1^2=81-20.25 \\ x1=\sqrt[]{60.75} \\ x1=7.79 \end{gathered}`
`\begin{gathered} x2 \\ 7.5^2+x2^2=15^2 \\ x2^2=225-56.25 \\ x2=\sqrt[]{168.75} \\ x2=12.99 \end{gathered}`

finally, we can find the area

`\begin{gathered} P1 \\ A1=(45\cdot7.79)/(2)=175.27 \\ P2 \\ A2=(75\cdot12.99)/(2)=487.12 \end{gathered}`

the ratio of their areas are:

`(A1)/(A2)=(175.27)/(487.12)=0.36\approx(9)/(25)`