Question

I need assistance with finding the area of shaded squares.
https://uttia2.owschools.com/media/g_geo_2015/8/groupi57.gif - Particular square in question.

Answer 1

It looks to me like perhaps a square minus a quarter circle

Answer 2

So would is be 6 squared minus the area of the quarter square?

Answer 3

That would be my guess

Answer 4

so 36-1/4 pi r^2

Answer 5

so 36 - .25 ^ pi * 2 * 6?

Answer 6

36-.25*pi*6*6

Answer 7

6^6?

Answer 8

No 6^2 or six squared

Answer 9

I get approx 7.73 for the answer

Answer 10

What would that be in an A = __ - _PI

Answer 11

ok 36-9 pi since .25*6*6 is 9

Answer 12

Alright, can you help me with another?

Answer 13

Sure

Answer 14

https://uttia2.owschools.com/media/g_geo_2015/8/group58b.gif

Answer 15

I need to find the area of the shaded square

Answer 16

Looks like (square minus quarter circle) times two to me.

Answer 17

Ahhh. I see it now. Alright I'll plug in the numbers. Give me one second.

Answer 18

If you need the same format looks like 8-2pi

Answer 19

-21.1327412287 this is what I got.

Answer 20

I get approx 1.72

Answer 21

How'd you get that?

Answer 22

ok one of those pieces is a square minus a quarter circle
so 2^2-.25*pi*2^2

Answer 23

That is 4-pi

Answer 24

There are two of those pieces so two times that result or 8-2pi

Answer 25

what about the pi*2*2?

Answer 26

don't forget it is a quarter circle so you need the .25 factor

Answer 27

Oh yeah.

Answer 28

https://uttia2.owschools.com/media/g_geo_2015/8/groupi56.gif and this one would be 33.333% of the triangle correct?

Answer 29

I'ts a triangle minus 3 fractional circles, trying to figure out the circle fractions now.

Answer 30

Ok it's 60 degrees so those are 1/6 circles

Answer 31

So triangle minus 3 1/6th circles or triangle - 1/2 circle

Answer 32

It would be 360 divided by 60 right? Because the angles would be central angles and the triangle would have 60, 60, 60 angles?

Answer 33

Yeah or rather 60/360 so 1/6 circles.

Answer 34

And lets see the area of the triangle would be 1/2 b*h with a base of 12 and a height of sqrt 27

Answer 35

Then would we multiply that by two?

Answer 36

which simplifies to 18 sqrt 3 for the triangle area

Answer 37

and times .5 so .5*12*sqrt 27

Answer 38

which reduces to 18 sqrt 3

Answer 39

For the area of the triangle?

Answer 40

Yes for the triangle area

Answer 41

I used pythagoras to figure out the height by the way.

Answer 42

Then would we do 18 sqrt 3 - .1666 (PI * 3 squared?)

Answer 43

close there are 3 1/6 circles so a net of a half circle

Answer 44

So 18 sqrt 3 - .5 (pi * 3 squared)?

Answer 45

yeah

Answer 46

or 18 sqrt 3 - 4.5 pi

Answer 47

Odd, I can't put in 18 sqrt 3. it give me _ sqrt _ instead of __ sqrt _

Answer 48

well we could factor 9 out of it.

Answer 49

so 2 sqrt 3 -.5 pi

Answer 50

It won't accept the 2 either.

Answer 51

Let me just double check the triangle area,

Answer 52

Okay.

Answer 53

oops base of 6 not 12

Answer 54

height is still sqrt 27 though

Answer 55

so halve the first part

Answer 56

9 sqrt 3 - 4.5 pi

Answer 57

Or 9 sqrt 3 9/2?

Answer 58

sure 9 sqrt (3) - 9/2 pi

Answer 59

\[9\sqrt{3}-\frac{ 9 }{ 2 } \pi\]

Answer 60

https://uttia2.owschools.com/media/g_geo_2015/8/group58a.gif

Answer 61

Ooh getting tricky still the shaded region?

Answer 62

Yes.

Answer 63

Hmm just to check, no calculus needed right.

Answer 64

Its geometry and I've learned only a little bit of trig, so I'd assume no.

Answer 65

Well the bottom is a half circle, the top is a lot trickier.

Answer 66

Should we start with the semi circle half then?

Answer 67

Sure that bit will be easy 1/2 pi r^2

Answer 68

so half pi 6 * 2?

Answer 69

sorry I meant 6 squared.

Answer 70

yeah or in other words 18 pi

Answer 71

Could we use ratios to find the other half?

Answer 72

Possibly what did you have in mind?

Answer 73

I think we may be able to get away with and I can't quite explain it that we have half a radius missing

Answer 74

The semi circles are only 6 high because of the radii and the non-shaded segment is only 3.

Answer 75

So we should (1/2)^2 of a circle area

Answer 76

or I think that might be a quarter circle of area on top.

Answer 77

If that was the case we would have 9 pi of area up there for a total of 27pi in shaded area.

Answer 78

I'll try that

Answer 79

er or maybe I should say a 1/4 of a half circle

Answer 80

in the A = _ pi - _ sqrt _ format would it be _ pi - 9 sqrt 3?

Answer 81

Hmm no sqrt involved so I guess this must not be right.

Answer 82

what would be the _ pi?

Answer 83

one possibity is 27

Answer 84

its only one space so it can't be

Answer 85

Just to confirm that dot is the center of the circle correct.

Answer 86

the 12? It doesn't say but I'd assume so.

Answer 87

Wait
I think I can put in 27.

Answer 88

Turns out there are two slots but it still won't accept 27

Answer 89

how about 22.5, or rather how many attempts do you get?

Answer 90

22.5 would be a 1/2 circle + 1/8 circle.

Answer 91

Given I only have 2 slots and not 4 I don't think I can do 22.5

Answer 92

Ok I've done some research looks like we need to draw some right triangles.

Answer 93

Oh boy!

Answer 94

Where does it go?

Answer 95

Hmm maybe not but it would go from the center to the edge and to the edge of the circle