I recently encountered this problem in a maths competition at school and even after days of working on it i just can't find any real way of solving it.

I've attached the problem.

I need to find the radius and the area of the shaded area

To me it really seems like it is a variable answer and has no set number, but i'm doubting that if so it would be in this quiz.

Any help????

 

What have you done so far?

 

The hint is to find calculate R^2 - r^2 where R is the radius of the large circle and r the radius of the small circle. Cut the chord in half using r and then relate this to R using Pythagoras. You can continue from there.

I think you need one more piece of infomation to solve this.

 

You need either the rise of the arc over the chord (h), or one of the radii ajb mentions R or r, or the angle subtended at the centre.

I think you need one more piece of infomation to solve this.

You only actually need R^2 - r^2 to solve this and you can calculate directly, you don't need R or r themselves.

Here is a spread sheet calculating the value of r given various values of h = R - r

 

 

 

Here is a spread sheet calculating the value of r given various values of h = R - r

 

chords1.jpg

 

 

Why is this spread sheet relevant? [latex]R^2-r^2=25.[/latex]

Edited August 5, 201510 yr by mathematic

Why is this spread sheet relevant? [latex]R^2-r^2=25.[/latex]

so does that mean the area is always constant irrespective of the radii values?

 

Why is this spread sheet relevant?

 

The question requires you to find r and R.

 

The relationship between them is h = R - r and R = h/2 + c²/8h where c is the chord length.

 

You are only given c and require one other of h, R or r to complete the calculation.

Edited August 5, 201510 yr by studiot

The question requires you to find r and R.

mathematic is right, assuming the circles are concentric and the chord is tangent to the inner circle; I think that is the case. You only need R^2 -r^2 to solve this problem. You just need a factor of pi now and the problem is solved.

 

post#1

 

I need to find the radius and the area of the shaded area

 

This is pretty clear to me.

 

Perhaps the OP was attempting the wrong problem.

I don't see how you can find the radius (of what exactly?) without extra information. Finding the area is easy enough.

Edited August 6, 201510 yr by ajb

 

I don't see how you can find the radius (of what exactly?) without extra information. Finding the area is easy enough.

 

 

Isn't that what I said in post#3

Isn't that what I said in post#3

Indeed, so we think the original question is ill posed. tom2326 has probabily misunderstood the question, or missed some vital information, like a value for r or R, depending on the exact question.

Yup.

 

My guess (after your responses) is that the original question only asked for the area, but Tom assumed you need to calculate the ( or a) radius and, of course, couldn't do that.

So he asked for the radius as well

 

I never considered that possibility in my original response.

Indeed, it is tempting to try to find R and r and then the areas and work out the difference. But that won't work!