Question

Find the surface area of the torus shown below, with radii r and r.

Answer 1

Final answer:

The formula for the surface area A of a torus, given its inner radius r and outer radius R, is:

Step-by-step explanation:

The surface area of a torus can be derived by considering its geometry. A torus is essentially formed by revolving a circle (with radius r) around a central axis at a distance R from the center of this circle, creating a "doughnut" shape.

To find the surface area of this shape, the formula for the surface area of a torus is . This formula can be understood in the following way:

- The term R represents the distance from the center of the torus to the center of the circular cross-section. This distance defines the "size" of the torus.

- The term r represents the radius of the circular cross-section itself.

- is a constant value that relates to the circular and rotational aspects of the torus' geometry.

The surface area of the torus is calculated by multiplying these values together, representing the area of the entire surface of the shape, including both the outer and inner surfaces of the "doughnut." This formula is a result of mathematical analysis and geometric considerations specific to the torus shape.

Answer 2

The formula for the surface area A of a torus, given its inner radius r and outer radius R, is:
`\[ A = 4 \pi^2 Rr \]`

The surface area of a torus can be derived by considering its geometry. A torus is essentially formed by revolving a circle (with radius r) around a central axis at a distance R from the center of this circle, creating a "doughnut" shape.

To find the surface area of this shape, the formula for the surface area of a torus is
`\(A = 4 \pi^2 Rr\)`. This formula can be understood in the following way:

- The term R represents the distance from the center of the torus to the center of the circular cross-section. This distance defines the "size" of the torus.

- The term r represents the radius of the circular cross-section itself.

-
`\(4 \pi^2\)` is a constant value that relates to the circular and rotational aspects of the torus' geometry.

The surface area of the torus is calculated by multiplying these values together, representing the area of the entire surface of the shape, including both the outer and inner surfaces of the "doughnut." This formula is a result of mathematical analysis and geometric considerations specific to the torus shape.

Question: