A solid has a circular base of radius 1. It has parallel cross-sections perpendicular to the base which are equilateral triangles. How do you find the volume of the solid?

Question

Samantha Adkison · Accepted Answer

#V = (4sqrt3)/3#

Set the origin of the axis at the center of the circle so that the origin is in the center and the parallel cross-sections have the base of the triangles parallel to the #y# axis.

For every #x in (-1,1)# the length of the base of the triangle will than be:

#b = 2sqrt(1-x^2)#

and since it is equilateral its height will be:

#h=sqrt(3)/2b = sqrt(3)sqrt(1-x^2)#

The area of the triangle is then:

#S = (bh)/2 = sqrt(3)(1-x^2)#

and the relative volume element:

#dV = sqrt(3)(1-x^2)dx#

Integrating over the interval:

#V = int_(-1)^1 sqrt(3)(1-x^2)dx = sqrt(3) [x-x^3/3]_(-1)^1#

#V = sqrt(3) (1-1/3+1-1/3) =(4sqrt(3))/3#

Evelyn Agard · Answer

undefined To find the volume of the solid, you can use the method of slicing perpendicular to the base and summing up the volumes of the individual slices. Since the cross-sections are equilateral triangles, you can express the area of each slice in terms of its height. Then, integrate the areas over the height of the solid to find the total volume.

The formula for the area of an equilateral triangle with side length $ s $ is $ A = \frac{\sqrt{3}}{4} s^2 $. In this case, the side length of each equilateral triangle is equal to the diameter of the circular base, which is 2.

Integrating this area formula over the height $ h $ of the solid, which ranges from 0 to the height of the solid, will give the total volume. So, the integral to find the volume $ V $ would be:

$$ V = \int_{0}^{h} \frac{\sqrt{3}}{4} (2)^2 \, dh $$

Solve this integral to find the volume of the solid.