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Expert Verified by: David Chen, M.Sc. Mathematics | Fact-checked by the BEP Math Team

Easily determine the volume of a solid of revolution using the disk or washer method. This professional calculator allows you to solve for volume, height, or radii by inputting your known geometric parameters.

Volume of a Solid Revolution Calculator

Result:
Please enter 3 variables to calculate the 4th.

Volume of a Solid Revolution Formula:

Washer Method:

$$V = \pi \times H \times (R_{out}^2 – R_{in}^2)$$

Source: Wolfram MathWorld, NASA PUMAS

Variables:

  • V (Volume): The total cubic space occupied by the generated solid.
  • H (Height): The height of the solid or the interval $[a, b]$ along the axis of revolution.
  • R_out (Outer Radius): The maximum distance from the axis of revolution to the curve.
  • R_in (Inner Radius): The distance from the axis to the internal boundary (0 for a solid object).

Related Calculators:

What is a Solid of Revolution?

A solid of revolution is a three-dimensional figure obtained by rotating a two-dimensional region around a straight line (the axis of revolution) that lies in the same plane. This concept is fundamental in integral calculus and engineering design.

Common examples include spheres (rotating a circle), cylinders (rotating a rectangle), and cones (rotating a triangle). Calculating the volume is essential for determining material requirements and weight in manufacturing.

How to Calculate (Example):

  1. Identify your axis of revolution (e.g., X-axis).
  2. Determine the outer radius ($R_{out} = 5$) and inner radius ($R_{in} = 2$).
  3. Measure the height or interval ($H = 10$).
  4. Apply the formula: $V = \pi \times 10 \times (5^2 – 2^2) = \pi \times 10 \times 21 \approx 659.73$.

Frequently Asked Questions (FAQ):

What is the difference between Disk and Washer methods? The Disk method is used for solid shapes ($R_{in} = 0$), while the Washer method accounts for a hollow center ($R_{in} > 0$).

Can the radius be negative? No, distance from the axis must be positive. If the function crosses the axis, the absolute value is typically used.

What if the rotation is around the Y-axis? The logic remains identical, but the height $H$ represents the change in $y$, and radii are functions of $y$.

Is the result in cubic units? Yes, the volume is always expressed in units cubed (e.g., $cm^3$, $in^3$).

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