The smaller square takes up one half of the larger square.

The diagonal of the smaller square equals the length of one side of the large square.

A = 1 side of the large square(Also the diagonal of the smaller square)
B = 1 side of the small square
X = Area of the large square
Y = Area of the small square

You can also easily figure it out without math, but the math way is more fun

Turn the smaller square at a 45 degree angle and draw a cross from the top corner of the smaller square to the bottom corner, and from the left corner to the right corner. Then you can easily see that it is 1/2 of the large square.

Just to add, rotate the inner square 45 degrees to form a diamond inside the larger one. Drawing two lines connecting opposite points of the smaller square (which cross at the centre of the circle), reveals four outer and four inner triangles of equal size. Ergo, one-half!

The small square covers one half of the large square. If you give each side of the small square a number, such as 1, and use the Pythagorean theorem to calculate the diameter of the circle, you will get one side of the big square. Square that number and you will get exactly twice the area as the small square.

The small square covers 1/4 of the big one, because from pitagoras theorem the square root of the sum of square of the sides of a triangle with 90° angle is equal to the square of the hipotenuse (third side) which happens to be the diameter of the circle and the side of the big square. So then each side of the small square is equal to diameter divided by square root of 2. Then multiplying (d/sqrt(2))^2 = d^2/2

the correct answer is 70.7 percent of the larger square. it is basic trig. the diameter of the circle equals the side of the larger square. the diagonal line in the smaller square equals the diameter of the circle. the constant for the hypotenuse of a 45degree right triangle is 1.414. 1 divided by 1.414 is .707 or 70.7 percent

i glommed the image into photoshop and measured the two squares. ok, you can call that cheating, but i was interested to know who’s right and who’s wrong. i’ll deal with “why” later.

the big square measures 200×200 pixels, or 40,000 sq. pixels.
the small square measures 140×140 pixels, or 19,600 sq. pixels.
(nb: 200*.707 ~= 140)

Lee| Guest January 25th, 2007 - 10:34 amThe smaller square takes up one half of the larger square.

The diagonal of the smaller square equals the length of one side of the large square.

A = 1 side of the large square(Also the diagonal of the smaller square)

B = 1 side of the small square

X = Area of the large square

Y = Area of the small square

A*A=X

B^2+B^2=A^2(Pythagorean Theorem)

B*B=Y

Therefore, 1/2X=Y

Lee| Guest January 25th, 2007 - 10:37 amYou can also easily figure it out without math, but the math way is more fun

Turn the smaller square at a 45 degree angle and draw a cross from the top corner of the smaller square to the bottom corner, and from the left corner to the right corner. Then you can easily see that it is 1/2 of the large square.

Vishvas Vasuki| Guest January 25th, 2007 - 11:05 amMy answer = half

-Vishvas Vasuki

Kathy| Guest January 25th, 2007 - 11:43 am1/2?

kira| Guest January 25th, 2007 - 12:33 pmlooks like about 4/9 to me (just eyeballing it)

andrew| Guest January 25th, 2007 - 2:24 pm50%

Rod Hines| Guest January 25th, 2007 - 3:40 pmOne-half!

Rod Hines| Guest January 25th, 2007 - 3:56 pmJust to add, rotate the inner square 45 degrees to form a diamond inside the larger one. Drawing two lines connecting opposite points of the smaller square (which cross at the centre of the circle), reveals four outer and four inner triangles of equal size. Ergo, one-half!

Josh| Guest January 25th, 2007 - 6:21 pmThe small square covers one half of the large square. If you give each side of the small square a number, such as 1, and use the Pythagorean theorem to calculate the diameter of the circle, you will get one side of the big square. Square that number and you will get exactly twice the area as the small square.

T'Surakmaat| Guest February 20th, 2007 - 11:35 amit is one-half. the sine of the diagonal is .5.

Carlos Lopez Silva| Profile February 24th, 2007 - 11:33 pmThe small square covers 1/4 of the big one, because from pitagoras theorem the square root of the sum of square of the sides of a triangle with 90° angle is equal to the square of the hipotenuse (third side) which happens to be the diameter of the circle and the side of the big square. So then each side of the small square is equal to diameter divided by square root of 2. Then multiplying (d/sqrt(2))^2 = d^2/2

dale| Guest April 29th, 2007 - 8:39 pmthe correct answer is 70.7 percent of the larger square. it is basic trig. the diameter of the circle equals the side of the larger square. the diagonal line in the smaller square equals the diameter of the circle. the constant for the hypotenuse of a 45degree right triangle is 1.414. 1 divided by 1.414 is .707 or 70.7 percent

T'Surakmaat| Guest April 29th, 2007 - 10:29 pmDale, you’ve forgotten to square .707.

.707*.707 ~= .4998, which is to say, 1/2.

i glommed the image into photoshop and measured the two squares. ok, you can call that cheating, but i was interested to know who’s right and who’s wrong. i’ll deal with “why” later.

the big square measures 200×200 pixels, or 40,000 sq. pixels.

the small square measures 140×140 pixels, or 19,600 sq. pixels.

(nb: 200*.707 ~= 140)

live long and prosper

T’Surakmaat

beth| Profile October 14th, 2008 - 2:32 pmi still dont unerstand it :S i get that its a half but i dont know how to work it out using algebra :s

michaelc| Profile October 14th, 2008 - 3:33 pmHi beth,

Let one side of the small square = s.

So area of the small square = s².

Now next step takes some insight! Diagonal of little square = diameter of the circle, which is the length of the sides of the big square.

So, pythagorean theorem is c² = a² + b².

In this case a=b=s, so

c² = 2s²

c = s?2

since c is the length of the sides of the big square,

c² is the area which is 2s².

s² to 2s², big square is twice the size or the little square is half the size, whichever is clearer.

Hope this helped.

beth| Profile October 17th, 2008 - 1:14 pmthnx i get it now lol