1. ## Sphere, Geometry Problem

A cone circumscribes a sphere and has its slant height equal to the diameter of its base.
Show that the volume of the cone is 9/4 the volume of the sphere.

Basically I need to show my professor that the volume of the cone is 9/4 the volume of the sphere.
Can someone help me with it or just give me like 3 solution steps so I can proceed. Thanks in advance.

2. ## Re: Sphere, Geometry Problem

If youcut the cone in two halves, you will get an equilateral triangle its vertices are vertices of an hexagon and its height is 3R/2 and R is the radius of the sphere. Continue from there

3. ## Re: Sphere, Geometry Problem

cone has an equilateral cross section side length d
volume of cone /volume sphere wiil cancel out to 9/4

4. ## Re: Sphere, Geometry Problem

I'm still confused. Is the radius of the sphere equal to the radius of the cone?

5. ## Re: Sphere, Geometry Problem

Originally Posted by PhillipMasagca
I'm still confused. Is the radius of the sphere equal to the radius of the cone?
No it is not. Use the information in the attached and use the properties of an equilateral triangle.

6. ## Re: Sphere, Geometry Problem

Originally Posted by PhillipMasagca
I'm still confused. Is the radius of the sphere equal to the radius of the cone?
look closely at the values I gave you for base radius and sphere radius and note they are not equal to each other

7. ## Re: Sphere, Geometry Problem

Originally Posted by bjhopper
look closely at the values I gave you for base radius and sphere radius and note they are not equal to each other
My apology. the center of an inscribed circle is the intersection point of the bisectors. In an equilateral triangle a bisector is also a median. The intersection point of the bisectors is also the intersection point of the medians. Therefore, in your problem the radius of the inscribed circle is 1/3 the length of the median, which is also the height of the equilateral triangle. You need to correlate the side of the triangle with the radius of the circle via the length of the height.

8. ## Re: Sphere, Geometry Problem

@votan after i get the radius of the sphere to be 1/3 of the height of the cone what is the next step. im kinda confused (Thanks Again)

9. ## Re: Sphere, Geometry Problem

Originally Posted by PhillipMasagca

@votan after i get the radius of the sphere to be 1/3 of the height of the cone what is the next step. im kinda confused (Thanks Again)

You got the radius of the sphere correctly assuming the height of the cone is one unit. but if you label the height of the cone h units, What the radius of the sphere would be.

Now you will need to relate h to the side of the equilateral triangle, which is the base of the triangle (why?)

10. ## Re: Sphere, Geometry Problem

Originally Posted by PhillipMasagca

@votan after i get the radius of the sphere to be 1/3 of the height of the cone what is the next step. im kinda confused (Thanks Again)

1/2 *d*rad3 = alt of cone
Both are properties of 30-60-90 triangles

11. ## Re: Sphere, Geometry Problem

Originally Posted by bjhopper
1/2 *d*rad3 = alt of cone
Both are properties of 30-60-90 triangles
Radius of sphere is third altitude of cone. The sphere is inscribed in the cone.

12. ## Re: Sphere, Geometry Problem

Originally Posted by votan
Radius of sphere is third altitude of cone. The sphere is inscribed in the cone.

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# a cone circumscribes a sphere and has its slant height equal to the diameter of its base. show the volume of a cone is 9/4 the volume of the sphere

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