# 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

• 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

# 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 $$4πr^2$$

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

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:

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