If you have, say, a cheesecake dish, which is 10 and a half inches in diameter, what...?

Question

is the maximum size SQUARE object which will fit inside it? The answer is somewhere near 7 and 1/4 inches but how would you calculate it?

Answer 1

Don't worry just make the cheesecake and save me a piece please !

Answer 2

Imagine the square inside the circle. They both share a common central point. Since a corner of the square touches the circle, and they both share a centre, the distance from the centre of the square to the perimiter of the circle is equal to the radius of the circle.

So, if the radius of the circle is R, the square will have a diagonal of 2R. Let the side of the square be S. From pythagoras:

    S^2 + S^2 = (2R)^2

So, 2S^2=4R^2,  and S= root ( 2 R^2)

For your example, S = root (2 * 5.25 * 5.25), which is about 7.4 inches

Answer 3

just give me the pie

Answer 4

Draw a square into a circle. Then you'll see that the square's diagonal is as big as the circle's diameter, 10.5 inches.

That's how I'd begin.... But I'm not good at maths, only in visuals :D LOL

Answer 5

times the area by pie..........

Answer 6

It's to do with the square root.

If you draw the square inside the circle you'll see that it's the square's corners which touch the circle. In other words, the diagonal size of the square is the same as the diameter of the circle. A 10.5 inch dish has a diameter of 10. 5inches, so the square has a diagonal of 10.5 inches. Put that fact on one side for a moment.

Pythagoras's theorem tells us that if we have a triangle with shorter sides of length x and y then the long side equals the square root of x squared plus y squared. Now go back to your square inside your circle. A square is the same as two triangles put together, so it should be obvious that the diagonal length of your square is equal to the square root of one side squared plus the other side squared.

Or, in maths,

   diag^2 = side1^2 + side2^2

We know side1 = side 2 (cos it's a square) and we know the diagonal is 10.5, so we have

    10.5 x 10.5 = 2 x side1^2

or

     110.25 = 2 x side1^2

or

      55.125 = side1^2

so

     side1 = square root of 55.125 = 7.425 (approx.)