Mathematicians - help on Y7 question!

[whenthewhistleblows]

How do I work out the area of the shaded part? It’s a follow up question from work on area of a circle.

Tia

Hmm.
There's probably a right way to do it. But I wouldn't know at my age.

The only way I can vaguely see that being possible is working it out as if that's circles inside boxes. (see my crap diagram)

So if I assumed that one of those arcs was from one circle with a radius of 6cm, then the full circle area would be 113.1 (pi x r squared)

Then I'd work out a quarter of that -> 28.28 which is shaded bit of circle plus one empty bit.
So if the area of the square is 36cm squared, then one of those empty bits is 7.72cm squared. (36-28.28)
Two of the empty bits is 15.44cm squared.
So the shaded bit is 36-15.44 = 20.56cm squared.
Thats probably completely nonsense. Best I can do.

I assume that this is one quarter of a circle, right?

If so, then you know that the area of the square is 36cm.

The area of a whole circle with a radius of 6cm is pi r2, so 3.14 x 62, so ~113cm.

If you then take 1/4 of that circle's area (28.25cm2), you know what the area of ONE of the unshaded parts is (36-28.25=7.75). Double this as there are two, and that means that the unshaded part is 15.5cm2, so the shaded part is 36-15.5 = 20.5cm^2.

Hope that's right!

Ah sorry, I see someone already answered! At least we seem to agree...

Area of square is 6^2 = 36.

Area of a quadrant with sides of length six is (pi.6^2)/4 = 9pi.

So the area of one of the unshaded bits is 36 - 9pi.

So the area of two of the unshaded bits is 72 - 18pi.

So the area of the shaded bit is 36 - 72 + 18pi = 18pi - 36.

The main thing I'd do differently to the solutions above, which give the same final answer, is work in terms of pi all the way through, because it's easier and more accurate.

Thinking out loud here...

Area of circles (assuming what we're seeing in the pic is a quarter of each circle) = 36pi
One quarter of the area of a circle = 9pi

Area of square = 36

Each white area = 36 - 9pi

So the shaded area is 36 - 2(36 - 9pi) = 20.54cm^2

I think?

Thank you - I’m going to have to print this out and get in ‘the zone’ to digest it as my brain has now melted!

God I love geometry. I haven't done any in so long.

Oh I think I get it!

You work out the area of the square
You work out the area of the circle quadrant
If you take away the area of the circle quadrant from the area of the square, you’ve got the area of one of the non shaded bits
Double that because there are two non shaded bits
Tske the sum of the two non shaded bits away from the area of the square and then you’re left with the area of the shaded bit

Is that right????

Yep - you got it!

Correct!

See picture

Thanks DadDadDad.

I feel like I need a cigarette! (I’ve never smoked).

So DadDadDads method (work out half the circle area then deduct area of two triangle areas) looks slightly different to the methodology I eventually got to grips with. I understand both (I think!) but does anyone know if there’s s particular way they teach it in Y7?

Well if you cut the ends off the shaded area, then rotated it in three dimensions about the long axis you would have a fat cigar shape so maybe you could smoke that.

So it is 20.5. or thereabouts. I can't believe I got it right in my rambling way LOL!

If I were doing it with Y7, I'd probably go with the earlier methods, which effectively looks at half of the area and recognises it comes from a quarter circle and half a square, find that area then double it.

The brighter pupils then might see that it's equivalent (as I've shown) to finding a half circle and taking off a whole square (albeit disguised as a larger triangle).

I think that is a hard question for y7, so I am expecting a higher ability set.
It isn't something DD1 was ever given (post GCSE now), or that DD2 (y9) has been faced with.
Any of the correct ways described will work, I doubt they teach 'here is the way to do it.' It is more flexibility of mind.

To be honest by ds’ maths teacher is a bit crap. He basically says ‘if there’s things you don’t know look on mymaths and learn it from there’

Ds didn’t know how to calculate the area or permitted of a circle (wasn’t taught it at primary) so I taught him the formulas and then we went to practice questions on mymaths and that came up.

As I said, ds teacher pretty useless at communication - I can’t get out of him what ds is supposed to know and not know.

Teen - you're right this is at the level of having to work out for yourself how to split this up rather than having a taught method for this specific shape. I think I did see this sort of thing occasionally in Higher GCSE papers when my DS was practising last year.

Ds didn’t know how to calculate the area or perimeter of a circle (wasn’t taught it at primary)

Shame. My kids were not only taught the perimeter (that the perimeter is pi.d is a matter of definition), but a rather lovely visualisation to derive the area. www.mathsisfun.com/geometry/circle-area-by-sectors.html There should be more of this sort of teaching: just knowing what the formula is doesn't really get you anywhere, but having a sense of why is much more useful.

Thanks for the link - I will go through that with him too :)