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# Surface Area and Volume of a Torus

"Torus" is the technical mathematical name for the three-dimensional shape of a doughnut. A torus is generated by rotating a ring about an axis outside of the ring. To find the surface area and volume of a doughnut, you need to know the outer radius **R** and the inner radius **r**, i.e., the radius of the doughnut hole. You can apply the formulas described below, or you can use the convenient torus calculator.

### Volume of a Torus

If the outer radius of the doughnut is**R**and the inner radius is

**r**, then the volume of the solid is given by the equation

Volume = (1/4)(pi

^{2})(R

^{2}- r

^{2})(R - r).

Recall that pi is the constant 3.14159265358...

### Surface Area of a Torus

The surface area formula for a torus isSurface Area = (pi)

^{2}(R

^{2}- r

^{2}).

**Example**

Suppose a bagel is 5 inches wide and the hole is 2 inches across. Then

**R**= 2.5 and

**r**= 1. The volume is thus

(1/4)(pi

^{2})(2.5

^{2}- 1

^{2})(2.5 - 1)

= 19.43 cubic inches.

The surface area is

(pi)

^{2}(2.5

^{2}- 1

^{2})

= 51.82 square inches.

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