Question

The polygons are regular polygons. Find the area of the shaded region to the nearest tenth.

Answer 1

@dmezzullo i need help

Answer 2

please help

Answer 3

We need one more constraint for this to be possible.

Answer 4

what do you mean this is all the problem is

Answer 5

Unless I'm missing something (which could be the case), there needs to be an extra constraint...

Answer 6

this is all it gave me

Answer 7

please help

Answer 8

I reckon I can have a crack if you still need help?

Answer 9

i need help

Answer 10

Ok, so I think we can break down the shapes in to two squares and 2 triangles.We know the length of the diagonal side, which is 4. With me so far?

Answer 11

still with you

Answer 12

Great, we also know that the sides of a square are all the same. Let's call them \(x\)Looking at the triangle with the sides I have marked x, we can use Pythagoras to calculate x. What is the value of x?

Answer 13

4

Answer 14

Nope use Pythagoras' theorem.\[a^2+b^2 = c^2\]In this case our value for c is 4. And our value for a and b is x, making our equation\[x^2+x^2 = 4^2\]Does that make sense?

Answer 15

yes

Answer 16

Ok, so what do you get for x?

Answer 17

2x^2=16 right

Answer 18

So far so good

Answer 19

then do you divide 2 to make

x^2=8

Answer 20

yep making x...?

Answer 21

x=64

Answer 22

No, square root both sides.
\[x^2=8 \rightarrow \sqrt{x^2}=\sqrt{8}\] Making \[x=\sqrt{8}\]

Answer 23

Oh, crap, missed the whole first part of the case "regular polygons."

Thanks, mate, for catching that!

Answer 24

No probs

Answer 25

Does that make sense @sandbeach ?

Answer 26

yes

Answer 27

Great now we know the length of the square, we can think of the shaded region as simply three identical squares with length \[\sqrt{8}\]. Do you know how to find the area of a square?

Answer 28

lenth x width

Answer 29

Awesome, so what is the area of one of these squares of length \(\sqrt{8}\)?

Answer 30

i dont know the anser to sqrt8 x 4

Answer 31

I have to go now, but basically the size of a square of length \(\sqrt{8}\) is 8. We know that there are 3 squares making up our shaded region.

\[3 \times 8 = 24\]
That is your answer.

Answer 32

so that is the area to the shded region