Presented here is a diagram of the inscribed cone in a sphere, where O is the center of the sphere:
The problem asks for us to determine the volume of both the cone and the sphere. The necessary component that is missing for the cone is the height from C. Notice that:
Our height is thus split into two components, determing the value of OA and OC.
The Quest for OCNotice how both OC and OB are given as radii of 5 cm. The fact that radii in a sphere are equal gives us:
The Quest for OAThe fact that AC is a perpendicular height, allows us to say that for
, the triangle has
Given that we know AB=4 from the problem, we can now apply Pythagorean Theorem.
Note: We could have done this faster using a Pythagorean Triple of 3-4-5, but I wished to show a general methodVolume of the ConeBy this, we have AC=3+5=8. The volume is now simple application of the volume of a cone.
Volume of the SphereApplication of the volume of a sphere.
Ratio of VolumesNotice how the pi and 1/3 cancels.
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