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Math Brain Teaser: The Unkindest Cut of All, Part 1 of 2

In honor of Mathematics Awareness Month, here is another mathematical brain bender from puzzle master Wes Carroll.

The Unkindest Cut of All, Part 1 of 2

Difficulty: HARD
Type: MATH (Spatial)

The area of a square is equal to the square of the length of one side. So, for example, a square with side length 3 has area (32), or 9. What is the area of a square whose diagonal is length 5?

In this puzzle you are working out your spatial visualization (occipital lobes), memory (temporal lobes), and hypothesis generation (frontal lobes).


I am especially fond of these two ways to solve this problem:

1. Draw the right triangle whose hypotenuse is the square’s diagonal, and whose two legs are two sides of the square. Then use the Pythagorean Theorem (a^2 + b^2 = c^2) to solve for the length of each side. Since two sides are equal, we get (a^2 + a^2 = c^2), or (2(a^2) = c^2) ). Since c is 5, 2(a^2) = 25, making a^2 equal to 25/2, or 12.5. Since the area of the square is a^2, we’re done: it’s 12.5.

2. Tilt the square 45 degrees and draw a square around it such the the corners of the original square just touch the middles of the sides of the new, larger square. The new square has sides each 5 units long (the diagonal of the smaller square), and it therefore has area 25. However, a closer inspection reveals that the area of the larger square must be exactly twice that of the smaller. Therefore the smaller square has area 25/2, or 12.5.

You can now go on to Concentric Shapes: The Unkindest Cut of All, Part 2 of 2


More brain teaser games:

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13 Responses

  1. Omar says:

    I feel the solution is too long, easiest thing to do is:

    you know that the diagonal is sq.rt(2)*x where x is the measurement of the side. since that’s true, then swuare the answer nd you’re done. so [5/2qrt(2)]^2 = 12.5

  2. Juan says:

    There is another easy way:

    The diagonal divides the square in two triangles. Each triangle has an area of bxh/2
    b = diagonal
    h = diagonal / 2

    so 5 x 2.5 / 2 = 6.25
    6.25 x 2 triangles = 12.5

    I think you would have to set the difficulty to Easy.


  3. Hymen says:

    I concur!

  4. Mitch says:

    a^2 + b^2 = c^2
    You square the length of the diagonal (C).
    25 = a^2 + b^2
    because its a square both sides are equal. So divide by 2 to get the square of 1 of the sides ie the volume.

  5. DJ_Dysfunk says:

    Notice two diagonal lines, one for each pair of opposing corners of the square, form four halves of 2 squares. The length of the side of any of these halves is equal to half of 5, or 2.5. The area of a square is one of its sides squared, 2.5^2 = 6.25 * 2 = 12.5, as (again) there are four halves of two squares.

  6. Sharon says:

    A^2+b^2=c^2. C = 5, and a =b. 5×5=25=c^2. A^2+b^2=25. a^2 and b^2 therefore both equal 12.5. A^2 is the formula for area, so the area is 12.5

  7. John says:

    12.53 where the square root of 2 = 1.41421356

  8. Egon_Freeman says:

    I concur, this puzzle’s difficulty setting should be changed to “Easy”. I arrived at the result in about 15 seconds, and that’s certainly not bragging…

  9. alside says:

    I believe the purpose for these puzzles is to exercise the portions of the brain that are in use when completing the activity.

  10. Bill says:

    You guys are making this too hard.

    Two isosceles triangles with hypotenuse 25.

    Each other side is the square root of 12.5.

    square root of 12.5 squared is 12.5

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