BIZOMAT MAKEOS · Geometry
Volume and Surface of a Cylinder Cut by a Plane (Ungula and Truncated Cylinder)
What is the volume and the curved surface area of a cylinder cut by a sloping plane, either through a chord of the base (an ungula or hoof) or across the whole cylinder between two heights?
Portions of a cylinder by the handbook’s formulas: for a cylinder cut by a plane through a chord of the base, V = (2/3 a³ ± b × base area) h ÷ (r ± b) and curved surface S = (a d ± b × arc length) h ÷ (r ± b), with a the half chord, b the distance from the axis to the chord, d the diameter, plus when the base is larger than half the circle and minus when smaller; and for a cylinder cut off at an angle across its full section, V = π r² (h1 + h2) ÷ 2 and S = π r (h1 + h2).
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How this tool works
Inputs, units and where to find them
| Input | Unit | Typical source |
|---|---|---|
| Cut | choice | Whether the plane cuts through a chord of the base or across the whole cylinder. |
| Cylinder diameter d | mm | Diameter of the cylinder. |
| Height of the cut at the top edge h (ungula) | mm | Height of the ungula at the point of the base furthest from the chord. |
| Distance from the axis to the chord b (ungula) | mm | Perpendicular distance from the cylinder axis to the chord where the plane meets the base. |
| Base of the ungula (ungula) | choice | Whether the base of the ungula is more or less than half the circle. |
| Longest side h1 (cut across) | mm | Height of the longest side of the cut-off cylinder. |
| Shortest side h2 (cut across) | mm | Height of the shortest side of the cut-off cylinder. |
What each input means
Open an input to see what it is, its units, the worked example and the common mistakes, with a small diagram.
What this means: Cut
- What it is
- Whether the cutting plane passes through a chord of the base or across the whole cylinder.
- Units and format
- Choose one of 2 options from the list: Plane through a chord of the base (ungula); Plane across the whole cylinder (heights h1 and h2).
- Example
- The form opens with “Plane through a chord of the base (ungula)” selected as a worked example. Choose the option that matches your case.
- Effect on the result
- Chooses the ungula or the cut-across formulas.
- Get it right
- A hoof or ungula starts at a chord of the base; a mitred pipe is cut across.
What this means: Cylinder diameter d
- What it is
- Outside diameter of the cylinder or round bar that is cut.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0.000001 mm to 100000 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 127 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Volume rises with its square.
- Get it right
- Measure across the round section square to the axis.
What this means: Height of the cut at the top edge h (ungula)
- What it is
- Height the cut reaches at the side of the base furthest from the chord.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0.000001 mm to 100000 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 50.8 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- The ungula volume and surface rise in proportion.
- Get it right
- Measure along the cylinder side from the base to the cutting plane.
What this means: Distance from the axis to the chord b (ungula)
- What it is
- Square distance from the cylinder axis to the chord where the plane meets the base.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0 mm to 100000 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 0 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Moving the chord changes the base area and the volume.
- Get it right
- Enter 0 when the chord passes through the centre of the base.
What this means: Base of the ungula (ungula)
- What it is
- Whether the base of the ungula covers more or less than half the circle.
- Units and format
- Choose one of 2 options from the list: Half the circle or less; More than half the circle.
- Example
- The form opens with “Half the circle or less” selected as a worked example. Choose the option that matches your case.
- Effect on the result
- Chooses the plus or minus signs of the formulas.
- Get it right
- The base is larger than half when the chord is on the far side of the axis from the high point.
What this means: Longest side h1 (cut across)
- What it is
- Height of the longest side of the cylinder cut off at an angle.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0 mm to 100000 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 150 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- The volume follows the mean of h1 and h2.
- Get it right
- Measure along the side where the cut is furthest from the base.
What this means: Shortest side h2 (cut across)
- What it is
- Height of the shortest side of the cylinder cut off at an angle.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0 mm to 100000 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 100 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- The volume follows the mean of h1 and h2.
- Get it right
- Measure along the side where the cut is nearest the base.
Method and formulas
- Ungula: a = √(r² − b²); segment area = r² acos(b ÷ r) − a b and arc = 2 r acos(b ÷ r) for a base of half the circle or less; for a larger base, area = π r² − that segment and arc = 2π r − that arc.
- V = (2/3 a³ ± b × base area) h ÷ (r ± b); S = (a d ± b × arc) h ÷ (r ± b); + for a base larger than half the circle, − for smaller.
- Cut across: V = π r² (h1 + h2) ÷ 2; S = π r (h1 + h2).
- Example: the volume is 136,560 mm³ and the curved surface 6,451.6 mm².
Limitations
- The plane must meet the base along the chord and rise to the height h at the far side of the base.
- For a pipe, subtract the volume of the bore cut the same way.
- Example values are illustrations. Replace them with values for your actual task.
Typical uses
- Find the volume of a hoof cut from a cylinder or round bar
- Work out the material in a pipe or bar cut at an angle
- Find the curved surface area of a mitred cylinder
Who it is for
- Fabricators
- Design engineers
- Estimators
What you get
- Base area and arc length of the ungula
- Volume
- Curved surface area
- Mass for a density, if wanted (multiply the volume)
Interpreting the result
- With the chord through the axis (b = 0) the ungula volume is 2/3 r² h.
- Cut across the whole cylinder, the volume equals a cylinder of the mean height.
References
- Machinery’s Handbook, 29th edition (Industrial Press, 2012), printed pp. 83–84. Topic reference; no source tables or recommended settings are reproduced.
Learn the Concepts Behind This Tool
These BIZOMAT Library readings explain the method this tool uses, with worked examples.
Useful alongside · Library collection
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In Practical Mathematics for Engineering and Manufacturing
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Open the Volume and Surface of a Cylinder Cut by a Plane (Ungula and Truncated Cylinder) calculator