Question

Can someone walk me through this problem :
A regular Pentagon has an Apothem of 3.2m and an area of 37.2m. What is the length of one side of the Pentagon?

Answer 1

I'm not completely sure but I think you have to multiply the length times width to get the area.

Answer 2

I found an equation for Area of a Regular Polygon :
Area = (½)nsr
n = # of sides
s = length of a side (What I'm trying to solve for)
and r = Apothem (3.2)

Answer 3

Is 4.67 an answer?

Answer 4

Area = (1/2)nsr *

Answer 5

It's multiple choice, and yeah

Answer 6

Maybe I miscalculated or something....I used the equation but I didn't get an answer that was a choice.
4.67 is an answer and I'm going to trust that it's correct.
I would like a walkthrough though...

Answer 7

First, do you know what the apothem is?

Answer 8

Actually, do you know how to find the radius given the apothem?

Answer 9

Yes, I know what the apothem is. And no I don't know how to find the radius given the apothem

Answer 10

Okay, the formula for the radius given the apothem is \(\dfrac a{2sin(\dfrac{180}{n})}\)

Answer 11

Okay

Answer 12

So substitute a and n in

Answer 13

The interior angles of a Pentagon would be...

Answer 14

Why do we need the interior angles?

Answer 15

Sum of interior angles = (n-2) * 180
= (5-2) * 180
= 3 * 180
= 540 degrees
540/5 = 108 degrees each angle

Answer 16

To find the sin in that equation you posted?

Answer 17

No

Answer 18

We already have a and n

Answer 19

Ohhhhhhhhhhhh -facepalm-

Answer 20

Okay so Radius = 37.2/2sin(180/3.2) ?

Answer 21

a is apothem, n is number of sides :)

Answer 22

R = 3.2/2sin(180/5)

Answer 23

Yes!

Answer 24

Okay :)

Answer 25

Sooooo

Answer 26

Ah shizzle, change that sin to cos

Answer 27

Okay

Answer 28

Hopefully you know about cosine?

Answer 29

R = 2

Answer 30

Yes COS = Opposite side/Hypotenuse

Answer 31

Okay, I trust that your math is correct :P

Answer 32

Okay so why did I need to find the radius?

Answer 33

Now we use a little trick called the Pythagorean Theorem

Answer 34

Okay....I think I see where this is going..It's so much better to visualize it!

Answer 35

We can see that the apothem cuts one side in half, and we are finding half of one side length, so once we find that, we can find the entire side.

Answer 36

Okay so I could do a(sq) + b(sq) = c(sq)?

Answer 37

Yep

Answer 38

10.24 - 4 = half of the side s (sq)

Answer 39

4.99

Answer 40

s = 4.99

Answer 41

Okay, so if 4.99 is half of one side, what is one side

Answer 42

No 4.99 is the full side

Answer 43

oh, okay :)

Answer 44

then you're done

Answer 45

Thank you very much :)

Answer 46

No problem :)

Answer 47

Could you help me on 1 more problem? I'll post it now

Answer 48

Sure :)

Answer 49

i have a different way of doing it
apothem is the inside radius
tan 54 =1.376=3.2/x
x=2.327
2x =side =4.65