This Lesson (OVERVIEW of LESSONS on Volume of SPHERES) was created by by ikleyn(52786): View Source, Show About ikleyn:
OVERVIEW of LESSONS on Volume of Spheres
For your convenience, this file contains
- the list of my lessons on volume of spheres in this site,
- the formula for calculating the volume of spheres, and
- the list of relevant solved problems.
Spheres
A sphere is a surface in a 3D space all points of which are equidistant from one point called the center of the sphere.
Figure 1a shows the sphere.
A radius of a sphere is a straight segment connecting the center of the sphere with a point on the sphere surface (Figure 1b).
A diameter of a sphere is the straight segment passing through the center of the sphere and connecting the opposite points of the sphere (Figure 1c).
Figure 1a. A sphere
Figure 1b. A sphere
and its radius
Figure 1c. A sphere
and its diameter
Properties of spheres
1. Every section of a sphere by a plane is a circle.
2. A tangent segment to a sphere released from a point in 3D space outside the sphere is perpendicular to the radius of the sphere drawn from its center
to the tangent point.
3. All tangent segments to a sphere released from one fixed point outside the sphere have the same length.
Formula for calculating the volume of spheres
The volume of a sphere is = = ,
where is the radius of the sphere and is the diameter of the sphere.
- Find the volume of a sphere if its radius is of 10 cm.
- Find the volume of a sphere if its radius is of 3 cm.
- Find the volume of a composite body comprised of a right circular cylinder and a hemisphere attached center-to-center
to one of the cylinder bases if both the cylinder diameter and the hemisphere diameter are of 20 cm, and the cylinder
height is of 40 cm.
- Find the volume of a composite body comprised of a right circular cylinder and a hemisphere attached center-to-center
to one of the cylinder bases if both the cylinder diameter and the hemisphere diameter are of 10 cm, and the cylinder
height is of 20 cm.
- Find the volume of a composite body comprised of a cone and a hemisphere attached center-to-center to the cone base
if both the cone base diameter and the hemisphere diameter are of 20 cm and the cone height is of 10 cm.
- Find the volume of a composite body comprised of a cone and a hemisphere attached center-to-center to the cone base
if both the cone base diameter and the hemisphere diameter are of 10 cm and the cone height is of 5 cm.
- Find the volume of a composite body comprised of a cube and a hemisphere attached center-to-center to one of the
cube faces if both the cube edge measure and the hemisphere diameter are of 20 cm.
- Find the volume of a composite body comprised of a cube and a hemisphere attached center-to-center to one of the
cube faces if both the cube edge measure and the hemisphere diameter are of 10 cm.
- Find the volume of the sphere inscribed in a cone if the base diameter of the cone is of 24 cm and the height of the
cone is of 16 cm.
- Find the volume of the sphere inscribed in a cone if the base diameter of the cone is of 30 cm and the height of the
cone is of 36 cm.
My lessons on volume of spheres and other 3D solid bodies in this site are