Solids With Known Cross Section

Solids with Known Cross Section

Cross sections help us visualize how multiple 2D planes can make up a 3D object. For example, a camping tent has cross sections with the shape of a triangle or rectangle (depending on if you are looking at the parallel or perpendicular angle) or a soccer ball has cross sections in the shape of a circle. The Egyptian Pyramids were even built one cross section square at a time. By breaking down a 3D object into simple 2D shapes, we can easily visualize a 3D model and calculate its volume.

For me personally, one of my favorite foods is ice cream and it’s to die for out of a waffle cone! But every time I go to an ice cream parlor, I struggle with the million-dollar question, waffle cone or cup? Off the top of my head, I choose a scoop in a cup because I think I’ll get more ice cream (which is the main goal of course!) but I just never know. To help figure this out, the function I chose to focus on is an absolute value function when rotated on the y-axis forms a cone with ellipse cross sections.

F(x) = -|5x|+1

After calculating the volume, I'll be able to tell just how much ice cream I'm getting out of a cone.

(For the purposes of math and 3D printing, calculations are scaled down and do not represent the volume of an ice cream cone. However, the same approach would work with correct measurements.)

Volume

Approximate Volume

To approximate the volume, I broke down the function into 10 elliptical cross sections. The cone has a height of 1 inch; therefore, each cylinder will be 0.1 inch tall. The cross sections and base of the cylinder dimensions are width of the absolute value and 2.5 of the absolute value.
To find the volume of each cross section we used the following formulas.
$A = \prod ab$

$V = A * height$

The approximate volume of my model is 0.12089 in³.

Actual Volume

To find the actual volume I integrated the area of the cross section in terms of y.

$V = \frac{5}{2}\prod \int_{-0.2}^{0.2}(\frac{1}{5}(1-y)^{2})$

$V = 0.12733 in^{3}$

These are similar which proves this an accurate way to calculate and visualize the solid. To double check my work I also calculated the volume using the volume of a cone equation. However, this will not be as accurate because this is for a more uniform shape. But I wanted to include it in this post for comparison reasons.

$V = \frac{1}{3}\prod abh$

$V = \frac{1}{3}\prod(0.2)(0.5)(1))$

$V = 0.1047 in^{3}$

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