Need to make a free standing cake slice shape

[kayona]

I feel really foolish and I'm not even sure this is the right forum. Here is what I need to make. I need to make a free standing cake slice shape. I know I need 3 squares for the sides and 2 triangles, one for the top and bottom . I need the squares to be 5 1/2 inches each. I need to know what the lengths of the sides of the triangle should be. I thought they would have to be 5 1/2 inches but when i cut them 2 sides fit and one is way too long........I know this is probably very easy, a math whiz i'm not........thank you for any help. Kayona

 

[soroban]

Hello, kayona!

I feel really foolish and I'm not even sure this is the right forum.
Here is what I need to make.
I need to make a free standing cake slice shape.
I know I need 3 squares for the sides and 2 triangles, one for the top and bottom.
I need the squares to be 5 1/2 inches each.
I need to know what the lengths of the sides of the triangle should be.
I thought they would have to be 5 1/2 inches but when i cut them 2 sides fit and one is way too long.

I guess I can't follow your description.
As I read it, there is no reason for your difficulty.

I think the shape looks like this:

              5.5
      * * * * * * * * * *
      | *           *   |
      |   *     *       |
  5.5 |     *           | 5.5
      |     |           |
      |     |           | 
      *     |           *
        *   |       *
     5.5  * |   *  5.5
            *

Aren't all the edges equal to 5.5 inches?
(You have three squares and two equilateral triangles.)

 

[mmm4444bot]

I know I need 3 squares for the sides

Can you say more about this? I mean, were you specifically told to use three identical squares?

but when i cut [the squares] 2 sides fit and one is way too long

Again, I would like more information. Two sides fit what, exactly? One side is way too long for what?

Is this an arts-and-craft project? Why are you building this wedge shape? What is the ultimate goal?

 

[soroban]

Hello, kayona!

Re-reading your problem, I think I see the difficulty.

How do you know your need three squares?
If they told you that, the problem is silly.

Since you say that the third piece is too big,
. . I suspect you left out a critical piece of information.
The cake slice is a specific fraction of the cake.


Suppose the slice is one-eighth of the cake.

Then the shape looks like this:

              5.5
      * * * * * * * * * *
      |:*.     45d  *   |
      |:::*.    *       |
  5.5 |:::::*           | 5.5
      |:::::|           |
      |:::::|           | 
      *:::::|           *
        *:::|       *
       x  *:|   *  5.5
            *

You have two isosceles triangles, two squares, and a rectangle.


Looking down on the piece, we have this diagram:

            *
           / \
          /45d\
     5.6 /     \ 5.5
        /       \
       /         \
      * - - - - - *
            x

\(\displaystyle \text{Law of Cosines: }\:x^2 \:=\5.5)^2 + (5.5)^2 - 2(5.5)(5.5)\cos45^o\)

. . . . . . . . . . . . . . \(\displaystyle =\;2(5.5)^2 - 2(5.5)^2\left(\tfrac{\sqrt{2}}{2}\right)\)

. . . . . . . . . . . . . . \(\displaystyle =\;2(5.5)^2\left(1 - \tfrac{\sqrt{2}}{2}\right)\)

. . . . . . . . . . . . . . \(\displaystyle =\; 2(5.5)^2\left(\tfrac{2-\sqrt{2}}{2}\right)\)

. . . . . . . . . . . . . . \(\displaystyle =\;(5.5)^2\left(2-\sqrt{2}\right)\)

\(\displaystyle \text{Therefore: }\:x \;=\;5.5\,\sqrt{2 - \sqrt{2}} \;=\;4.209517756 \;\approx\;4.2\text{ inches}\)