Dish End Volume Calculator: Formula & Examples

This calculator works out the internal volume of a dished end (also called a dished head) used on pressure vessels, storage tanks, and boilers. You enter the dish’s inside diameter, the depth (height) of the dish, and the type of dish shape (elliptical, torispherical, or hemispherical), and the calculator returns the volume in cubic meters or liters. It uses the standard geometric formulas for each head shape, so you get a reliable figure for partial or full liquid capacity.

Formula

The volume of a dished end depends on its shape. For a hemispherical head, the volume is exactly half a sphere: V = (π × d³) / 12, where d is the inside diameter. For an elliptical head (typically 2:1 ellipsoidal), the volume is V = (π × d² × h) / 6, where h is the dish depth. For a torispherical head (like a flanged and dished head), the volume is approximately V = 0.44 × (π × d³) / 12, which is a common empirical factor used in tank design. In all formulas, d and h must be in the same units (meters, inches, etc.), and the result is in cubic units of that measurement.

Worked Example

Using the default values: inside diameter d = 2.0 m, dish depth h = 0.5 m, and elliptical type. For an elliptical head, the volume formula is V = (π × d² × h) / 6. Plug in the numbers: V = (3.1416 × (2.0)² × 0.5) / 6 = (3.1416 × 4.0 × 0.5) / 6 = 6.2832 / 6 = 1.0472 m³. The calculator shows 1.047 m³, which rounds to 1.047. In liters, that is 1,047 L. If you had chosen hemispherical, the volume would be (π × 2³) / 12 = (3.1416 × 8) / 12 = 2.094 m³, exactly double the elliptical result for a 2:1 head because the depth is half the diameter.

When to Use This Calculator

Use this calculator when you need the capacity of a dished end alone, for example to determine the total volume of a horizontal or vertical vessel by adding the cylindrical shell volume plus two dished ends. It is also useful for estimating the amount of liquid that fills a dished head during hydrostatic testing, or for calculating the volume of material needed to form the head (if you multiply by wall thickness, though that is an approximation). The calculator is accurate for the internal volume based on inside dimensions, which is what matters for liquid capacity.

How to Choose the Right Dish Type

Selecting the correct dish type is critical because the volume differs significantly:

• Hemispherical: Deepest and strongest, but uses the most material. Volume is the largest for a given diameter.
• Elliptical (2:1): Common in pressure vessels, depth is 25% of the diameter (for 2:1 ellipsoidal, h = d/4). Volume is moderate.
• Torispherical: Shallower, cheaper to fabricate, but weaker. Volume is about 44% of a hemisphere, which is close to a 2:1 ellipse but slightly less.

If you are unsure, check the vessel drawing or standard (e.g., ASME VIII) to see the head type and dimensions. The depth h is usually measured from the tangent line (where the head meets the shell) to the center of the dish.

Common Mistakes and Misconceptions

One common mistake is using the outside diameter instead of the inside diameter. The volume of liquid or gas inside the head depends on the inside dimensions, so always use the inside diameter and inside depth. Another error is mixing units—if the diameter is in meters, the depth must also be in meters, otherwise the result is off by a factor of 1000 or more. Also, do not confuse the dish depth with the straight flange length (the cylindrical section at the edge of the head). The straight flange is not part of the dished volume; it is part of the shell.

A frequent misconception is that a torispherical head has exactly half the volume of a hemisphere. In reality, it is about 44% of a hemisphere, not 50%. The difference matters for large tanks. Also, the formula for elliptical heads assumes a true ellipsoid; some heads are “flanged and dished� which are torispherical, not elliptical, so verify the shape.

Limitations of the Formula

These formulas assume a perfect geometric shape with uniform thickness. Real heads have manufacturing tolerances, so the actual volume may vary by a few percent. The torispherical factor of 0.44 is an approximation; the exact volume depends on the knuckle radius and crown radius, which vary by standard. For precise engineering, use the actual head drawing or a CAD model. Also, the calculator gives the internal volume of the head alone—not the total vessel volume. To get the total, add the volume of the cylindrical shell (π × r² × length) and any other heads.

Frequently Asked Questions

What is the difference between a dished end and a dish end?

They are the same thing. “Dished end� is the more formal term used in engineering, while “dish end� is a common shorthand. Both refer to the curved cap that closes the end of a cylindrical pressure vessel or tank.

How do I measure the dish depth?

Measure from the tangent line (the point where the curved dish meets the straight cylindrical shell) to the center of the dish’s interior surface. This is the inside depth h. Do not include the straight flange length, which is the short cylindrical extension used for welding.

Can I use this for a vertical tank with only one dished end?

Yes. If the tank has one flat bottom and one dished top, the total volume is the cylindrical section plus the dished end. If both ends are dished, add both. This calculator gives the volume of a single dished end, so you can add it as needed.

Why does the elliptical head volume not use the depth in the formula?

Actually, it does. For a 2:1 elliptical head, the depth is fixed at d/4, so the formula V = (π × d² × h) / 6 simplifies to V = (π × d³) / 24 when h = d/4. But if your head has a different depth (e.g., 1.5:1 or 2.5:1), you must use the general formula with the actual h. The calculator above uses the general formula for elliptical, so you can input any h.

What if my dish is not a standard shape?

If the head is a custom shape, you cannot use these simple formulas. You would need to calculate the volume by integration or use a 3D CAD model. For standard heads, these formulas are accurate enough for most purposes.

Does the calculator account for wall thickness?

No, it uses inside dimensions. If you have outside dimensions and wall thickness, subtract twice the thickness from the outside diameter to get the inside diameter, and similarly for the depth, before entering values. For thin walls, the difference is small, but for thick heads it matters.

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in d | Inside diameter | m | 2.0 | 0
in h | Dish depth (inside) | m | 0.5 | 0
sel type | Dish type | Elliptical=0.5, Torispherical=0.44, Hemispherical=0.5
out vol | Dish volume | m³ | (pi * d^3 / 12) * (type == 0.5 ? 1 : (type == 0.44 ? 0.44 : 1)) | 3
out vol_l | Volume in liters | L | vol * 1000 | 0