What's the area of this volume?

In summary, we have a 3d coordinate system with axis x,y and z and a circle on the x-z plane with function z^2 + (x-a)^2 = a^2. We rotate this circle 90 degrees around the z-axis and find the volume of the resulting surface. By using Cavalieri's principle and the area function for a given angle, we get V=\int_{0}^{\frac{\pi}{2}}\pi{a}^{2}ad\theta, which simplifies to V=\frac{\pi^2}{2}a^3, making our final answer a quarter of a torus.
  • #1
Jin314159
Consider a 3d coordinate system with axis x,y and z.

We are given a circle on the x-z plane with function [tex]z^2 + (x-a)^2 = a^2[/tex]. We rotate this circle 90 degrees around the z-axis. What's the volume of the resulting surface?
 
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  • #2
Well, shouldn't that be a quarter of a torus?
I'll opt for that and say:
[tex]V=\pi{a}^{2}*\frac{\pi}{2}a=\frac{\pi^{2}}{2}a^{3}[/tex]

Edit:
The area function for a given angle measured relative to the x-axis (and with the origin as the pole) is [tex]f(\theta)=\pi{a}^{2}[/tex]

By Cavalieri's principle, we have:
[tex]V=\int_{0}^{\frac{\pi}{2}}\pi{a}^{2}ad\theta[/tex]
 
Last edited:
  • #3


To find the volume of the resulting surface, we first need to determine the shape of the surface. From the given information, we know that the original circle lies on the x-z plane and has a radius of a. When rotated 90 degrees around the z-axis, this circle will form a cylinder with a height of a and a radius of a.

Using the formula for the volume of a cylinder, V = πr^2h, we can calculate the volume of the resulting surface as V = πa^2(a) = πa^3.

Therefore, the volume of the resulting surface is πa^3, where a is the radius of the original circle.
 

What is the formula for finding the area of a volume?

The formula for finding the area of a volume is dependent on the shape of the object. For a cube or rectangular prism, the formula is A = l * w * h, where l is the length, w is the width, and h is the height. For a cylinder, the formula is A = 2 * π * r * h, where r is the radius and h is the height. For a sphere, the formula is A = 4 * π * r^2, where r is the radius.

Can I use the same formula for finding the area of any 3D object?

No, the formula for finding the area of a volume is dependent on the shape of the object. Different shapes require different formulas to find the area. It is important to correctly identify the shape of the object before attempting to find its area.

Do I need to know the volume of an object to find its area?

No, the volume and area of an object are two different measurements. The volume is the amount of space an object occupies, while the area is the measure of the surface of an object. It is possible to find the area of an object without knowing its volume.

How can I measure the area of an irregularly shaped object?

The area of an irregularly shaped object can be measured by breaking it down into smaller, regular shapes. Find the area of each regular shape and then add them together to get the total area of the irregular object.

What units should I use when measuring the area of a volume?

The units used to measure the area of a volume will depend on the units used to measure the length, width, and height of the object. It is important to keep the units consistent throughout the calculation to get an accurate measurement of the area.

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