Sphere, Ellipsoid, and Torus Help

Introduction to the Sphere, Ellipsoid, and Torus (Donut)

There exists an incredible variety of geometric solids that have curved surfaces throughout. Here, we’ll look at three of the most common: the sphere , the ellipsoid , and the torus .

Surface Area and Volume of a Sphere

The Sphere

Consider a specific point P in 3D space. The surface of a sphere S consists of the set of all points at a specific distance or radius r from point P . The interior of sphere S , including the surface, consists of the set of all points whose distance from point P is less than or equal to r . The interior of sphere S , not including the surface, consists of the set of all points whose distance from P is less than r .

Surface Area Of Sphere

Imagine a sphere S having radius r as shown in Fig. 8-11. The surface area, A , of the sphere is given by:

A = 4π r ²

Fig. 8-11. A sphere.

Volume Of Sphere

Imagine a sphere S as defined above and in Fig. 8-11. The volume, V , of the solid enclosed by the sphere is given by:

V = 4π r ³ /3

This volume applies to the interior of sphere S , either including the surface or not including it, because the surface has zero volume.

Surface Area and Volume of an Ellipsoid

The Ellipsoid

Let E be a set of points that forms a closed surface. Then E is an ellipsoid if and only if, for any plane X that intersects E, the intersection between E and X is either a single point, a circle, or an ellipse.

Figure 8-12 shows an ellipsoid E with center point P and radii r ₁ , r ₂ , and r ₃ as specified in a 3D rectangular coordinate system with P at the origin. If r ₁ , r ₂ , and r ₃ are all equal, then E is a sphere, which is a special case of the ellipsoid.

Fig. 8-12. An ellipsoid.

Volume Of Ellipsoid

Imagine an ellipsoid whose semi-axes are r ₁ , r ₂ , and r ₃ (Fig. 8-12). The volume, V , of the enclosed solid is given by:

V = 4π r ₁ r ₂ r ₃ /3

Surface Area and Volume of a Torus (Donut)

The Torus

Imagine a ray PQ , and a small circle C centered on point Q whose radius is less than half of the distance between points P and Q . Suppose ray PQ , along with the small circle C centered at point Q , is rotated around its end point, P , so that point Q describes a circle that lies in a plane perpendicular to the small circle C . The resulting set of points in 3D space, “traced out” by circle C , is a torus.

Figure 8-13 shows a torus T thus constructed, with center point P . The inside radius is r ₁ and the outside radius is r ₂ . The torus is sometimes informally called a “donut.”

Fig. 8-13 . A torus, also called a “donut.”

Surface Area Of Torus

Imagine a torus with an inner radius of r ₁ and an outer radius of r ₂ as shown in Fig. 8-13. The surface area, A , of the torus is given by:

A = π ² ( r ₂ + r ₁ )( r ₂ − r ₁ )

Volume Of Torus

Let T be a torus as defined above and in Fig. 8-13. The volume, V , of the enclosed solid is given by:

V = π ² ( r ₂ + r ₁ )( r ₂ − r ₁ ) ² /4

Sphere, Ellipsoid, and Torus Practice Problems