How do I find the area of a square which a circle cannot reach?

Question

I have a square (of any size) with the largest circle inside it that will fit inside that square. How do I deternmine the percentage or fraction of the portion outside of the circle? I also need to explain/validate my findings? This is my son's assignment and I am trying to determine if he is on track or not, before he goes in front of his class to solve. I was a poor math student myself, so I can really use your help. Any help or ideas are appreciated.

Answer 1

(4-π)/4
{for a picture, see: http://mudandmuck.com/str2/inscribedCircle.jpg }
If the radius of the circle is R, then the area of the circle is πR^2
The square will then have a side length of 2R.
The area of this square is (2R)^2 = 4R^2
The area outside the circle is then 4R^2 - πR^2
Then the ratio of this outside area to the area of the square is:
(4R^2 - πR^2)/(4R^2) = (4-π)/4

Answer 2

We'll find the area of the square, subtract the area of the circle, and then divide this result by the area of the square.

Suppose the length of the side of the square is S. (Don't worry, we'll cancel away this variable in the end.) The area of that square is S^2.

The radius of the circle is the distance from the center of the circle to an edge of the square; this is S/2. The area of a circle is pi*r^2, so the area of this circle is pi*(S/2)^2, which is equal to pi*(S^2)/4.

The area outside the circle is therefore S^2 - pi*(S^2)/4. Factoring the S^2, this is (S^2)(1 - pi/4).

Finally, we want the ratio of (area outside the circle) / (total area of the square), which is (S^2)(1 - pi/4) / (S^2). This simplifies to 1 - pi/4.

Answer 3

if the circle is the largest that can fit in the square then it will touch the square at four points (one on each side).

therefore the diameter of the circle will equal the length of a side of the square

multiply that number by itself to get the area of the square

for the portion outside the square take the area of the square and subtract the area of the circle (area of a circle = the radius squared times pi). this will give you the area outside the circle, but inside the square. divide that number by the area of the square for the fraction of the portion outside the circle.

Answer 4

The side length of the square is the diameter; therefore, the area of the square would be d^2 and the area of the circle would be pi*d^2/4. The difference between these two is (4d^2-pi*d^2)/4 = d^2(4-Pi)/4.

Answer 5

OK.
You have a circle inside a square with the circle touching the edges of the box from the inside.

So...

The diameter of the circle is the same as the length of one side of the square.

The radius of the circle is (1/2)* side.
The area of the circle is pi* [(1/2) * side]^2 = (pi/4)*(side^2)

The area of the square is (side^2).

Now, the area of the square that is outside the circle is the difference (subtract): side^2 - [(pi/4)*(side^2)]

And, the fraction of the square that is outside the circle is:

{side^2 - [(pi/4)*(side^2)]} / side^2 =

1- (pi/4)

Thus, the percentage is 100% * (1 - pi/4)

Done!

Answer 6

Note that the circle inside the square is called the inscribed circle. If you draw it, you will note that the radius of the circle is exactly half the length of the side of the square. Therefore, we have:

Area of circle = pi*(s/2)^2 = pi*s^2 / 4
Area of square = s^2

So Area of parts not in circle: s^2 - pi * s^2 / 4 =

s^2 * (4-pi) / 4

So the percentage is (4-pi) / 4, which is approx. 21.5 %

Answer 7

the maximum size of the circle will have a diameter the same length of one of the sides of the square, we will call this a.

area of a circle = pi * (d/2) ^ 2
= pi * (a/2) ^ 2
= pi * (a^2) / 4
area of a square = (side length)^2 = a^2

percentage of the square the circle occupies will be:

% = (area of circle) / (area of square)
= (pi * (a^2) / 4) / (a^2)
= pi / 4
~= 78.54%

then the percentage that it is not occupying is:
100% - 78.54% = 21.46%

Answer 8

Statements approximately infinity are continually dodgy. it incredibly is a sturdy approximation to assert that a circle is a polygon with a limiteless form of corners. Technically, a circle is the shrink of a many sided widely used polygon through fact the form of components (and form of corners) will advance in direction of infinity. despite if it incredibly is merely an approximation, in that a circle does not have any corners or immediately components.