Geometry Problem: Show Area & Volume of Sphere & Cylinder

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Deduce from the previous problem, and the known volume of the cone, that the volume of the sphere is 2/3 the volume of the circumscribing cylinderUsing the result from the previous problem, calculate the volume of the sphere and compare it to the volume of the circumscribing cylinder. You will find that the volume of the sphere is 2/3 the volume of the cylinder.
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ber8
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Guys please help me with this:

1) Show that the slice z=c of the sphere x² + y² + z² =1 has the same area as the slice z=c of the cylinder x² + y² =1 outside the cone x² + y² = z².

2) Deduce from the previous problem, and the known volume of the cone, that the volume of the sphere is 2/3 the volume of the circumscribing cylinder

Thanks in advance
 
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  • #2
ber8 said:
Guys please help me with this:

1) Show that the slice z=c of the sphere x² + y² + z² =1 has the same area as the slice z=c of the cylinder x² + y² =1 outside the cone x² + y² = z².

Calculate the areas of those two shapes, then show that their equations are equal. To find the area between the cone and cylinder, find them both and subtract.
 

1. How do I calculate the area of a sphere?

The formula for calculating the area of a sphere is A = 4πr², where r is the radius of the sphere. First, square the radius and then multiply it by 4π to find the area.

2. What is the formula for finding the volume of a sphere?

The formula for calculating the volume of a sphere is V = (4/3)πr³, where r is the radius of the sphere. Cube the radius, multiply it by 4/3, and then multiply it by π to find the volume.

3. How can I show the area of a cylinder?

The formula for calculating the area of a cylinder is A = 2πr² + 2πrh, where r is the radius of the base and h is the height of the cylinder. First, calculate the area of the circular base by using the formula A = πr². Then, multiply the circumference of the base (2πr) by the height (h) and add it to the area of the base to find the total area of the cylinder.

4. What is the formula for finding the volume of a cylinder?

The formula for calculating the volume of a cylinder is V = πr²h, where r is the radius of the base and h is the height of the cylinder. First, calculate the area of the circular base by using the formula A = πr². Then, multiply the area of the base by the height to find the volume of the cylinder.

5. Can I use the same formula for calculating the area and volume of a sphere and a cylinder?

No, the formulas for calculating the area and volume of a sphere and a cylinder are different. However, the formula for finding the volume of a cylinder can be derived from the formula for finding the volume of a sphere by multiplying it by the height of the cylinder.

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