Volume & Surface Area of a Torus (Ring)

Visual, interactive torus calculator with full derivation. Enter inner radius \( r_1 \) and outer radius \( r_2 \).

Torus Calculator

Radius values ( \( r_2 \ge r_1 \) ) – results in cubic & square units.
TORUS DIMENSIONS
PRECISION

Results

— units³
— units²

* Surface area is the lateral (ring) area, not including the two circular "ends" (which are zero for a torus).

Geometry of the Torus

A torus (ring) is generated by rotating a small circle of diameter \( d \) along a larger circle of radius \( R \) (distance from center of tube to center of torus).

Torus ring 3D
Figure 1: Torus ring – inner radius \( r_1 \), outer radius \( r_2 \)

From the diagram: \( r_2 = R + d/2 \), \( r_1 = R - d/2 \). Adding gives \( r_1 + r_2 = 2R \), subtracting gives \( r_2 - r_1 = d \).

Torus cut along xy and yz planes
Figure 2: Cuts of the torus – inner (\( r_1 \)) and outer (\( r_2 \)) radii

Imagine slicing the torus and unrolling it into a cylinder (Figure 3). The cylinder has diameter \( d = r_2 - r_1 \) and length equal to the circumference of the large circle: \( 2\pi R = \pi (r_1 + r_2) \).

Torus developed as a cylinder
Figure 3: Torus "unwrapped" – a cylinder of diameter \( d \) and length \( 2\pi R \)

Volume Formula

Volume of the cylinder = area of base × height = \( \pi (d/2)^2 \times (2\pi R) \). Substitute \( d = r_2 - r_1 \) and \( R = (r_1+r_2)/2 \):

\[ V = \pi \left( \frac{r_2 - r_1}{2} \right)^2 \times 2\pi \left( \frac{r_1 + r_2}{2} \right) = \frac{1}{4}\pi^2 (r_2 - r_1)^2 (r_1 + r_2) \]

Lateral Surface Area

Lateral area of the cylinder = circumference of base × height = \( (\pi d) \times (2\pi R) \). Substitute \( d \) and \( R \):

\[ A_L = \pi (r_2 - r_1) \times \pi (r_1 + r_2) = \pi^2 (r_2 - r_1)(r_1 + r_2) \]

✅ Final formulas used in calculator ✅
\( V = \frac{\pi^2}{4}(r_2-r_1)^2(r_1+r_2) \)   |   \( A_L = \pi^2 (r_2^2-r_1^2) \)

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