I obviously can't use the answer to work it out and I thought that I would have to find the area of the smaller sector and minusing it from the bigger sector but I don't have theta... arghh, help please?
The area of a segment is equal to theta/2pi * area of the whole circle. Knowing that, you can work out the areas of the shaded segments in terms of theta, set them equal, and solve.
What I'm saying is that I have no idea how to go about showing that theta equals that value :/
You need to write down all the facts you have at your disposal. So:
LeftShadedArea = RightShadedArea LeftShadedArea + RightNonshadedArea = Area of small circle RightNonshadedArea + RightShadedArea = Area of large sector
Can you convert these into equations using the radii that you are given and the unknown angle theta?
Ooh, wow. My maths teacher always insists that you cannot. Maybe it depends on the exam board? We do OCR MEI, who do you mark for?
OCR. Your maths teacher is teaching you proper mathematics - it is bad form to work backwards but we try to award marks where we can so starting with the angle and finishing with equal areas would receive full credit.
OCR. Your maths teacher is teaching you proper mathematics - it is bad form to work backwards but we try to award marks where we can so starting with the angle and finishing with equal areas would receive full credit.
I can sort of see your reasoning, but I would not tend to publicize this. Although the angle-to-area argument is reversible in this case, I can see all sorts of dangers for people who don't understand how if-and-only-if conditions should be tested in general.
I can sort of see your reasoning, but I would not tend to publicize this. Although the angle-to-area argument is reversible in this case, I can see all sorts of dangers for people who don't understand how if-and-only-if conditions should be tested in general.
Don't worry, I have no plans for a major newspaper ad campaign.
The area of a segment is equal to theta/2pi * area of the whole circle. Knowing that, you can work out the areas of the shaded segments in terms of theta, set them equal, and solve.
You need to write down all the facts you have at your disposal. So:
LeftShadedArea = RightShadedArea LeftShadedArea + RightNonshadedArea = Area of small circle RightNonshadedArea + RightShadedArea = Area of large sector
Can you convert these into equations using the radii that you are given and the unknown angle theta?
This is what I ended up doing (on the left-hand side):
that's fine.
In terms of the language I used in my earlier post, this is what I did: LeftShaded + RightNonShaded = area of small circle = pi(r^2) = 16pi (using r=4) (i) RightNonShaded + RightShaded = sector area = (1/2)R^2(theta) = 18theta (using R = 6) (ii) But RightShaded = LeftShaded so we also have: RightNonShaded + LeftShaded = 18theta (iii)
Comparing (i) and (iii) tells us that 16pi = 18theta, or theta = (16pi/18) = 8pi/9.