Question

Find the area of the shaded region in the figure where α = π/3, b = 19. (Round your answer correct to the nearest whole number).

Answer 1

how do you find the segment thats not shaded?

Answer 2

but you know that an triangle with angle 60 degree is an equilateral triangle using SAS property

do you know it ?

Answer 3

from what result that the 3rd side is equal 19 too

Answer 4

ok i get it so you would subtract the area of the segment and the area of the circle right? or is that wrong

Answer 5

no

can you calcule area of this circle with radius =19 ?

Answer 6

yeah area= pie*r^2 for the circle

Answer 7

yes

so than will be how many

Answer 8

?

Answer 9

pie*19^2 which is equal to 361pi

Answer 10

yes so than now you need calculi area of this triangle equilateral

do you know it how ?

Answer 11

i thought you're supposed to find the area of the unshaded part?
than you subtract that from the area of the circle?

Answer 12

yes is right on this way too if you can calculi the area of this arc

but how you calculi it ?

Answer 13

so can you calculi area of this triangle ?

Answer 14

yeah calculate the area of the segment thats unshaded
wouldnt that be 19 as well or no?

Answer 15

sorry but i dont understand you

Answer 16

okay so what i mean is subtract the area of the circle and the area that is unshaded

Answer 17

its kind of hard to explain how im thinking but what would you do?

Answer 18

so my opinion is that you can calculi easy area of this circle like first step
2.step calculi area of this triangle equilateral

3. you know that this alpha angle has 60 degree so than you can calculi area of this part of circle without this 60 degree part and after you need just assuming this with area of this equilateral triangle and so will get the area of this unsheded part

Answer 19

do you understand me ?

Answer 20

i dont understand steps 2 and 3

Answer 21

so after you get area of this circle you can rewriting that

to 360 degree ------ is area 361pi
so to 360-60 degree will be ---- x
-------------------------------------
x = 300*361 /360pi =

Answer 22

so in this way you will get area of this circle without this 60 degree part

Answer 23

will get area of this circle without part AOB

ok ?

Answer 24

but from this part area of this equilateral triangle is shaded again too

so for this you need assuming to this resulted area of semi circle area of triangle

ok ?

Answer 25

i really dont understand your method..too complicated lolll thank you though

Answer 26

@TeenaMathew The one?

Answer 27

@e.mccormick yes

Answer 28

OK. Did they teach you how to find the area of an arc segment?

Answer 29

the area of equilateral triangle is s^2x sqrt(3)/4
hence s=19
A=156.317585383

Answer 30

well i learned 2 years ago loll i dont its 1/2*r right?

Answer 31

sorry i dont remember..when i found the area of the segment i got 180.5 is that correct?

Answer 32

Sorry bout that. Connection issues. OK.
Now, if you think about it, you can find any sector becaue it is part of a circle. If the area of the circle is \(pi r^2\) then the area of any sector is part of that. A segment, in contrast, is the little bit on the outer edge when you slice it with a line. Well, you now have a sector minus the traingle part to get the area of the sector. However, they seem to want everything but the sector, if I am reading that right. So you find the secotr, then take the area of the entire circle and subtract the area of the sector from that!

Answer 33

so wouldnt the area of the sector be 180.5 and the area of the circle is 361 pie

Answer 34

Yah, the circle is \(361\pi\). The sector is \[\frac{1}{2}\alpha r^2\] So \[\frac{1}{2}\cdot\frac{\pi}{3}\cdot 361\]

Answer 35

And I am working this as I go, so it takes a little time. hehe.

Answer 36

yeah i did the same that and then i stopped thinking that was it loll

Answer 37

in the sector formula what does a stand for?

Answer 38

The problem was given with an angle of alpha, so I am using that. Just like using \(\theta\)
Now, I can subract the area of the triangle to get just the segment, or I can use the formula for a segment. Someone else did it the first way, so I'll do the second.
\[\frac{1}{2}(\alpha-\sin\alpha)r^2\]

Answer 39

i got 10673.68 which doesnt look like the right answer lol

Answer 40

\[\frac{1}{2}\left(\frac{\pi}{3}-\frac{\sqrt3}{2}\right)361\] is the segment and nothing but the segment. So we could take the entire circle, and subtract just this segment, and that would be the shaded area because pages are usually white and shading is non-white.
\[361\pi-\left(\frac{1}{2}\left(\frac{\pi}{3}-\frac{\sqrt3}{2}\right)361\right)\]

Answer 41

i now got 9539.57

Answer 42

is that correct or wrong maybe a miscalculation

Answer 43

I got \(\approx 32.70\) for the segment, so \(992.74-32.70=960.04\)

Answer 44

where did you get the 32.70?

Answer 45

From the segment formula.

Answer 46

okay thank you! you are a life saver!