An octagonal shaped cake

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Question from Sara, a student:

Hi, I have tried many polynomial equations and substitution methods
but i cant seem to get the right answer. Can you please provide some
guidance?
Question:
A square cake tin is 24cm x 24cm x 7cm but a cook wants to make an
octagonal shaped cake so
has cut out 4 pieces of rectangular card board and placed one on each
corner on the interior of the tin.
The cake has to be a regular octagon so pieces must be just right.
What is the size of each piece of cardboard to make the cake a
regular octagon?
Thankyou.

Hello Sara,

Let’s try to draw a picture of this situation.

octagon

We will forget about the depth of the pan for now to simplify the picture since it makes no difference.
Since the octagon is regular, all of its sides must be equal. Can we find formulae for the lengths shown by the double sided arrows above?

Well, the sides of the square are 24 cm, so if we call the length of the sides of the triangles cut out of the cake to be, x, then the side of the octagon is 24 - 2x. (Why?)

We also know that the triangle is a right triangle (how?), so we can use the Pythagorean theorem to determine the length of the hypotenuse.

Since the octagon is regular, these formulae must be equal, so set them equal and solve for x.
What are the dimensions of the 4 cardboard pieces?

Tyler

Math Central is supported by the University of Regina and The Pacific Institute for the Mathematical Sciences.