Geometry

Spheres

Big Picture

Spheres can be thought of as 3-dimensional circles. While all the points of a circle are equally far from the center in a 2-dimensional space, the equidistant points of spheres are in 3-dimensional space.

Key Terms

Sphere: The set of all points in three-dimensional space that are equidistant (equally far) from a point.

Characteristics of a Shpere

A sphere is a 3-dimensional circle. Think of it like a ball.

  • A sphere is the set of all points that lie a fixed distance r from a center point.
  • A sphere is the surface that results when a circle is rotated about any of its diameters.
  • A sphere results when you construct a polyhedron with an infinite number of faces that are infinitely small.

Parts of a Sphere

Parts of a Sphere
  • Center: the point at the center of a sphere; all points on the sphere lie equidistant from the center
  • Diameter: length of segment connecting any two points on the sphere’s surface and passing through the center
  • Radius: the length of a segment connecting the center of the sphere with any point on the sphere’s surface
  • The radius is 1⁄2 the diameter
  • Secant: line, ray, or line segment that intersects a circle or sphere in two places and extends outside of the circle or sphere
  • Tangent: intersects the circle or sphere at only one point
  • All tangents are perpendicular to the radii that intersect with them
  • Great circle: plane that contains the diameter
  • The great circle is the largest circle cross section
  • There are infinitely many great circles
  • Hemisphere: half a sphere
  • A great circle divides a sphere into two congruent hemispheres
circle divides a sphere

Geometry

Spheres Cont.

Surface Area and Volume of a Sphere

Theorem: The surface area of a sphere is \(SA = 4πr^2\), where r is the radius.

To remember the formula for the surface area of a sphere, think of a baseball. The cover of the baseball can be approximated by four circles. Each circle has an area of \(πr^2\), so the surface area of the baseball is \(4πr^2\)

Surface Area of a Sphere
Surface Area  of a Sphere

Image Credit: Baseball image copyright Shebeko, 2014. Used under license from Shutterstock.com.

Theorem: The volume of a sphere is \(V = \frac{4}{3} \pi r^3\),  where r is the radius.

We can approximate the volume of the sphere by adding up the volumes of an  infinite number of infinitely small pyramids, as illustrated below

The volume of a sphere

Each of the faces is the base of a pyramid with the vertex located at the center. Since the volume of the pyramid is \(\frac{1}{3}Bh\), the volume of the sphere is

\(V_{all pyramids} = V_1 + V_2 + V_3 + ...  V_n\)

\( = \frac{1}{3}(B_1h + B_2h + B_3h + ... B_nh\)

\( = \frac{1}{3}h(B_1 + B_2 + B_3 + ... B_n\)

The sum of all the bases is the surface area of the sphere. The formula for the volume of a sphere can then be simplified as:

\(V = \frac{1}{3}h(4 \pi r^2) = \frac{4}{3} \pi r^3\)

Notes