# Torus Calculator

Calculate the surface area and volume of a torus based on its radius and tube radius.

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The Torus Calculator is a tool that enables you to calculate the surface area and volume of a torus, which has the shape of a doughnut. A torus is a surface of revolution created by rotating a circle in three-dimensional space around a line that does not intersect the circle.

To use the calculator, input the radius of the tube (r) and the radius of the torus (R). The calculator will then compute the surface area and volume of the torus using Pappus's centroid theorems.

, where s is for the arc length of the curve being rotated, d is for the distance traveled by the centroid of the region in one rotation,

, where A is for area of the region being rotated, d is for the distance traveled by the centroid of the region in one rotation,

The surface area of the torus is given by the formula

This formula represents the product of the circumference of the circle formed by rotating the tube and the circumference of the circle formed by the rotation of that circle.

The volume of the torus is calculated using the formula . This formula represents the product of the area of the circle formed by rotating the tube and the circumference of that circle.

There are three different classes of standard tori based on the relative sizes of R and r. When R > r, it is referred to as a standard ring torus. The case R = r represents a horn torus, which is a torus with no hole. Finally, when R < r, it describes a self-intersecting surface known as a spindle torus.

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