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# Sphere, Ellipsoid, and Torus Help

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## 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 .

### 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 2r 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 2r 1 ) 2 /4

## Sphere, Ellipsoid, and Torus Practice Problems

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