Dish Radius Calculator

The Dish Radius Calculator calculates the radius of a circular dish from its depth and diameter using basic geometry.

About the Dish Radius Calculator

This tool finds the radius of the sphere that would produce your dish if you cut out a cap from it. Many bowls, skylight domes, and pressure lids are spherical segments. If you know the rim-to-bottom depth and the rim diameter, you can compute the sphere radius with a compact formula.

The calculator focuses on a circular, symmetric profile. It treats your dish as a “spherical cap.” From your inputs, it reports the sphere radius, the center’s offset above the rim plane, and optional area or volume values. This keeps layout work, jig building, and quality checks fast and repeatable.

Use it during concept sketches, shop-floor checks, or field measurements. You’ll see the exact steps, a worked example, and caveats so your measurement approach matches the math behind the result.

How the Dish Radius Method Works

The method uses classical circle geometry. Picture a cross-section through the dish’s symmetry axis. The rim forms a chord of a circle, and the dish depth is the sagitta (the height from the chord to the arc). From those two values, the sphere radius is fixed.

• Model the dish as a spherical cap (a cut from a sphere).
• Measure the rim diameter (chord) and the dish depth (sagitta).
• Apply the sagitta–chord radius formula to compute the sphere radius.
• Compute the center offset as radius minus depth to locate the sphere center.
• Optionally compute area and volume if you need material or load estimates.

Because the cross-section is a circle, you only need two measurements. This is fast, robust, and easy to verify with a quick back-check against the original measurements.

Formulas for Dish Radius

Here are the core formulas behind the calculator. They assume a circular rim and a spherical profile. Use consistent units throughout.

• Radius from diameter and depth: R = (D²)/(8d) + d/2, where D is rim diameter and d is depth (sagitta).
• Radius from chord (c) and sagitta (d): R = (c²)/(8d) + d/2. Here c = D if the chord spans the entire rim.
• Depth from radius and diameter: d = R − sqrt(R² − (D/2)²).
• Diameter from radius and depth: D = 2 · sqrt(2Rd − d²).
• Spherical cap surface area: A = 2πRd.
• Spherical cap volume: V = (π d²/3) · (3R − d).

Worked example: A dish with D = 300 mm and d = 30 mm has R = 300²/(8 · 30) + 30/2 = 90,000/240 + 15 = 375 + 15 = 390 mm. The sphere center sits R − d = 360 mm above the rim plane. You can confirm by plugging R and D back into the depth formula.

What You Need to Use the Dish Radius Calculator

Gather a few measurements and choices before you start. This keeps your session quick and your result reliable.

• Rim diameter D (or chord length c) measured across the dish’s opening.
• Dish depth d (sagitta) from the rim plane down to the deepest point.
• Unit selection (mm, cm, m, in, ft) for inputs and outputs.
• Profile type: spherical (default) or parabolic (to also show focal length).
• Optional tolerance or measurement repeat count for averaging.

Depth must be positive. Very small depths relative to diameter produce very large radii, which can magnify rounding noise. If the rim is not truly circular or the dish is not rotationally symmetric, expect the formula to fit poorly. For shallow dishes, take care with depth measurement accuracy.

Using the Dish Radius Calculator: A Walkthrough

Here’s a concise overview before we dive into the key points:

1. Select your measurement units.
2. Measure the rim diameter across the widest opening and enter it.
3. Measure the dish depth from the rim plane to the lowest point and enter it.
4. Choose “Spherical” as the profile type unless you specifically need parabolic data.
5. Click Calculate to compute the radius and related values.
6. Review the result, then optionally copy, export, or print your calculation for records.

These points provide quick orientation—use them alongside the full explanations in this page.

Case Studies

Wood bowl layout: A woodturner checks a salad bowl blank. Measurements: diameter D = 300 mm, depth d = 30 mm. Applying the formula, R = 300²/(8 · 30) + 30/2 = 375 + 15 = 390 mm. The sphere center is R − d = 360 mm above the rim plane. What this means: Set your trammel or template to a 390 mm radius to match the intended curvature.

Architectural dome: A circular skylight opening is D = 2.4 m, depth d = 0.4 m. Compute R = 2.4²/(8 · 0.4) + 0.4/2 = 5.76/3.2 + 0.2 = 1.8 + 0.2 = 2.0 m. The center is 1.6 m above the rim plane. What this means: Structural and glazing checks can reference a 2 m sphere, simplifying panel layout and load modeling.

Assumptions, Caveats & Edge Cases

This method assumes a true circular rim and a spherical interior surface. Real-world parts sometimes differ. Keep these points in mind when you interpret the result and choose a formula.

• Non-spherical profiles (parabolic, elliptical, torispherical) require different formulas.
• Out-of-round rims or warped dishes can distort depth readings and inflate error.
• Very shallow dishes (tiny d) amplify measurement noise and rounding effects.
• If there is a step, lip, or fillet near the rim, measure depth from the true plane.
• Flexible materials can sag under their own weight; support and measure consistently.

If your use case is a torispherical or elliptical head, check the appropriate design code. Those shapes often specify a “crown radius” and a “knuckle radius,” not a single radius. The spherical-cap formula here will not model that geometry accurately.

Units & Conversions

Consistent units are critical. Mixing inches and millimetres leads to large errors, especially for shallow dishes. Use the same unit for diameter and depth, and the calculator will return a coherent result.

To use the table, convert your measurement to a common base (meters or inches), apply the formula, and convert back if needed. This avoids mixing units mid-calculation and keeps rounding effects predictable.

Troubleshooting

If your output looks odd, start by checking measurement and units. Most issues come from a swapped diameter/radius, a sagging tape line, or mixed units. Re-measure the depth with a straightedge across the rim to ensure a true rim plane.

• Result seems too large: depth may be too small or units mixed.
• Result seems too small: you may have entered radius instead of diameter.
• Inconsistent readings: average several depth measurements at rotated angles.

When parts are not symmetric, two measurements at 90 degrees can reveal ovality. If they disagree, the spherical-cap model will only approximate your shape.

FAQ about Dish Radius Calculator

Is dish radius the same as the bowl’s rim radius?

No. The dish radius here is the radius of the parent sphere, not the rim’s half-diameter. The rim radius is D/2, while the sphere radius comes from the formula using depth and diameter.

Can I compute the radius if I only know three points on the arc?

Yes, but you must first derive the chord and sagitta from those points. The perpendicular bisector method finds the circle center and radius. Then you can verify with the sagitta formula.

What if my dish is parabolic rather than spherical?

Use the parabolic focal length formula f = D²/(16d) for optics or antennas. A parabola does not have a single “sphere radius,” so do not apply the spherical-cap formula to it.

How accurate is the result for very shallow dishes?

Shallow dishes are sensitive to small errors in depth. A 1 mm error can swing the computed radius a lot. Use precise tools, take multiple readings, and average them.

Dish Radius Terms & Definitions

Dish radius

The radius of the sphere that would generate the dish if the dish were a spherical cap.

Diameter (D)

The straight-line distance across the rim opening, measured through the dish’s center.

Sagitta (depth, d)

The vertical distance from the rim plane to the deepest point of the dish.

Chord

A straight line cutting across a circle; here, it is the rim line spanning the opening.

Spherical cap

A slice of a sphere cut by a plane; its geometry sets the area, volume, and radius relations.

Center offset

The distance from the rim plane to the sphere’s center, equal to R − d for a spherical cap.

Parabolic dish

A surface shaped by a parabola rotated around its axis; characterized by focal length, not a sphere radius.

Torispherical head

A pressure-vessel end with a spherical crown and a smaller knuckle radius; not modeled by a single spherical-cap formula.

Sources & Further Reading

Here’s a concise overview before we dive into the key points:

• Wolfram MathWorld: Sagitta of a Circle
• Wikipedia: Spherical Cap — Definitions and Formulas
• HyperPhysics: Circular Segment Relations
• Engineering Toolbox: Spherical Segments
• Wikipedia: Parabolic Reflector and Focal Length
• Wikipedia: Pressure Vessel Heads Overview

These points provide quick orientation—use them alongside the full explanations in this page.