Question

Awarding medal for best answer.
When BA = 10 ft, find the area of the region that is NOT shaded. Round to the nearest whole number.
A. 52 ft^2
B. 305 ft^2
C. 43 ft^2
D. 262 ft^2

Answer 1

We'll start by working out the area of the circle; as BA is the radius, the area of the circle is 10*10*pi=100pi. The angles are opposite => congruent. So the interior angle of the sector is 60, which is equal to 60/360=1/6 of the whole circle, so the area of the sector is approximately equal to 16.7pi, now we need to find the area of the segment and subtract it from the area of the sector

Answer 2

To find the area of the segment we'll need to use the formula ½ × ( (θ × π/180) - sin θ) × r^2

Answer 3

@Dahlioz where did this formula come from: ½ × ( (θ × π/180) - sin θ) × r^2 ?

Answer 4

So after substituting the values of the angles and radius, etc, we get this;.5((60*3.14/180)-0.866025403784)*100, which is equal to 9.0320631441333333

Answer 5

woah wut

Answer 6

that's not even an option -.-

Answer 7

Not done, we need to subtract this from the area of the sector; 16.7pi-9.0320631441333333= 52.(3)-9.0320631441333333, which is equal to 43.3012701892, which we then round to 43

Answer 8

@agent0smith I have experience working with this myself, so I know the formula, though I'll find you some references anyway

Answer 9

@Dahlioz i try to avoid posting formulas like that, that a student wouldn't have access to themselves in class. You need to show them how to derive it, by making a triangle inside the sector of the circle, and subtracting the area of the triangle from the sector.

Answer 10

oh hahaha @Dahlioz thank you

Answer 11

We need to find the area of that triangle. The 60 degree angle will be divided in half, then we can find the height h, from cos30 = h/10, and the base from sin30 = b/10 (note that b is only half the base)

Then use Area = 1/2 base * height to find the area of that triangle.

THEN you have to subtract the area of the triangle from the area of that 60 degree sector.

And finally, subtract that result from the area of the circle.

Answer 12

You're right, @agent0smith , that would be a more pedagogical way of showing it, though using the formula as well can be more efficient and quick, as long as you know why you use it and how you can derive it in case you forget it

Answer 13

But as you've already shown it, it's not really necessary for me to do it anymore. Thanks! :)

On a side note, in this paragraph (the last sentence): "Then use Area = 1/2 base * height to find the area of that triangle.

THEN you have to subtract the area of the triangle from the area of that 60 degree sector.

And finally, subtract that result from the area of the circle.", Don't you mean "subtract that result from the area of the circle sector" ?

Answer 14

No, we're finding the area of the unshaded region, so we subtract the area of the triangle from the sector (to get the area of the shaded region), then subtract the area of the shaded region from the circle.

Answer 15

Or you could add the area of the triangle to the area of the remaining 300 degree sector of the circle, to get the same result.

Answer 16

Ahh, I tend to do stupid mistakes when I'm tired

Answer 17

Thanks for that, made me realize that I was wrong, the correct answer would be 314-9.0320631441333333=304.9679368558666667, which we'd round to 305

Answer 18

I initially thought what we were trying to find was the unshaded area of the sector rather than the unshaded area of the circle. I'll blame it on lack of sleep :)