The volume of the slice is 40 in³. The volume of the remaining cake is 197.014 in³.
A regular hexagon is defined as a closed shape consisting of six equal sides and six equal angles.
Here we have two regular hexagons
one top small hexagon cake with side length = 3 in, height = 3 in
One big hexagon cake, side length = 4 in, Height = 4 in
A slice cut such that it removes a side segment is equivalent to an equilateral triangle with side length = length of hexagon side
Also, all angles within the equilateral triangle are 60° each
Therefore, the length of the side of the removed equilateral triangle side
Top small cake slice triangle side = 3 in.
Area of surface of small slice = 1/2 x b x h = 1/2 x 3 x 3 x sin 60
= 9√3/ 4
The volume of a small slice
= Area of surface small slice × Height of small cake
= 9√3/ 4 x 3
= 11. 69
Big cake slice triangle side = 4 in.
Area of the surface of big slice = 1/2 x 4 x 4 x sin 60
= 4√3
The volume of big slice = Area of surface of big slice × Height of big slice
= 16√3
= 28
Total volume of slice = Volume of small slice + Volume of big slice
Total volume of slice = 12 in³ +28 in³ = 40 in³
For the small cake, the remaining volume = 5 x 11.69 = 58.45
For the big cake the remaining volume = 5 x 27.71 = 138.56
Total volume remaining cake = 58.45 in³ + 138.56 in³ = 197.014 in³
a = Length of side
h = Height of hexagon
The volume of each slice is, therefore,
= a^2 x h x √3/4
For the small cake, we have
a = 3 in.
h = 3 in.
Volume of small slice = a^2 x h x √3/4
= 9 x 3 x √3/4 = 27√3/4
For the big cake, we have
a = 4 in.
h = 4 in.
Volume of big slice = a^2 x h x √3/4
= 16 √3
The total volume of slice = Volume of small slice + Volume of a big slice
Total volume of slice = 27√3/4 + 16 √3
The total volume of slice = 39404 in³.
Learn more about hexagon;
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