# Sphere, Geometry Problem

• Sep 21st 2013, 06:37 AM
PhillipMasagca
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.
• Sep 21st 2013, 09:46 AM
votan
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
• Sep 21st 2013, 01:56 PM
bjhopper
Re: Sphere, Geometry Problem
cone has an equilateral cross section side length d
volume of cone /volume sphere wiil cancel out to 9/4
• Sep 21st 2013, 05:04 PM
PhillipMasagca
Re: Sphere, Geometry Problem
I'm still confused. Is the radius of the sphere equal to the radius of the cone?
Thanks for replying i appreciate your feedbacks
• Sep 21st 2013, 06:46 PM
votan
Re: Sphere, Geometry Problem
Quote:

Originally Posted by PhillipMasagca
I'm still confused. Is the radius of the sphere equal to the radius of the cone?
Thanks for replying i appreciate your feedbacks

No it is not. Use the information in the attached and use the properties of an equilateral triangle.
Attachment 29228
• Sep 22nd 2013, 06:23 AM
bjhopper
Re: Sphere, Geometry Problem
Quote:

Originally Posted by PhillipMasagca
I'm still confused. Is the radius of the sphere equal to the radius of the cone?
Thanks for replying i appreciate your feedbacks

look closely at the values I gave you for base radius and sphere radius and note they are not equal to each other
• Sep 22nd 2013, 09:37 AM
votan
Re: Sphere, Geometry Problem
Quote:

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.

Attachment 29234
• Sep 24th 2013, 04:00 AM
PhillipMasagca
Re: Sphere, Geometry Problem
@bjhopper what is 1/2drad3? is it 1/2(diameter)(radius^3)?? (Thanks Again)

@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)
• Sep 24th 2013, 05:59 AM
votan
Re: Sphere, Geometry Problem
Quote:

Originally Posted by PhillipMasagca
@bjhopper what is 1/2drad3? is it 1/2(diameter)(radius^3)?? (Thanks Again)

@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?)
• Sep 24th 2013, 07:35 AM
bjhopper
Re: Sphere, Geometry Problem
Quote:

Originally Posted by PhillipMasagca
@bjhopper what is 1/2drad3? is it 1/2(diameter)(radius^3)?? (Thanks Again)

@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
1/2 *d /rad3 = radius of sphere
Both are properties of 30-60-90 triangles
• Sep 24th 2013, 06:44 PM
votan
Re: Sphere, Geometry Problem
Quote:

Originally Posted by bjhopper
1/2 *d*rad3 = alt of cone
1/2 *d /rad3 = radius of sphere
Both are properties of 30-60-90 triangles

Radius of sphere is third altitude of cone. The sphere is inscribed in the cone.
• Sep 24th 2013, 08:24 PM
bjhopper
Re: Sphere, Geometry Problem
Quote:

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

1/2*d*1/rad3 = radius of sphere = 1/3 of alt of cone