Let the radius of the circular pie be r and the thickness t. Also let the number of slices be N. We can then find a formula for the dimensions of the container. The length of the isosceles triangle forming the base and lid of the container is the height of the triangle, which is equal to r. The angle of the isosceles triangle is 360/N.
To find the length of the isosceles triangle's base, drop a perpendicular from the apex of the triangle on to its base. The perpendicular will bisect the apex angle, so we have two back to back right-angled triangles with one angle equal to 360/2N=180/N. The half-length of the base of the triangle=rtan(180/N). So the base of the triangle is 2rtan(180/N). We then need to decide how many slices we are going to put into the container. Let S be the number of slices. So the height of the container is St at least to accommodate the slices.
Now we can calculate the area of the prism. It has 3 rectangular sides and 2 triangles for top and bottom. The long side of the container has length L given by r/L=cos(180/N) so L=r/cos(180/N).
Rectangular areas: StL=rSt/cos(180/N) (two of these), base * thickness=2rtan(180/N)*St=2rSttan(180/N).
Triangular base and lid: r^2tan(180/N) (two of these).
The total area is the sum of these 5 shapes: 2r(St(sec(180/N)+tan(180/N))+2rtan(180/N)).
If the container is to hold the entire pie then S=N.
Now put in some figures: r=23cm, t=0.8cm, N=S=6. 180/N=30. tan30=0.5774, cos30=0.8660. St=4.8cm.
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