Question

Dante wants to create a new school crest. He will use two identical regular pentagons--one green, on top of one black. He will cut square out of the green pentagon so that the black pentagon shows through.

a) Each pentagon can be cut from a 2 m by 2 m square piece of felt fabric. What area of felt will Dante buy?

b) What is the area of the black pentagon?

Area of regular pentagon:

c) How much green felt is wasted?​

Answer 1

a)

well, we know each pentagon can be cut from a 2x2 piece, so hmm Dante wants two pentagons, that's just two 2x2 pieces, or two 4m² of felt, 8m² of felt.

b)

Let's recall, the black pentagon is the one behind, Dante never cut the black one, only the green one, so we can say the area of the black one is the area of a pentagon with an apothem of 1.0m and a side of 1.5m.

`\textit{Area of a Regular Polygon}\\\\ A=\cfrac{ap}{2} ~~ \begin{cases} a=apothem\\ p=perimeter\\[-0.5em] \hrulefill\\ a=1\\ p=\stackrel{1.5* 5}{7.5} \end{cases}\implies A=\cfrac{(1)(7.5)}{2}\implies \stackrel{ \textit{black pentagon} }{A=3.75~m^2}`

c)

how much felt is wasted? well, Dante cut out a 40cm x 40cm square in the green pentagon, so that got wasted, that'd be 40x40 or 1600 cm².