Volume Calculation for a Truncated Cone 1. Vertical Truncation °
Author: J007BONDExecutions: 1334
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Find the volume V of the portion on the right side of the cross-section (the shaded area) when a right circular cone with radius R and base angle δ is cut vertically (parallel to the central axis).
The cutting position is expressed as an angle Θ [°]. It can be any value within the full range (0°–180°).
The cutting angle γ is fixed at 90 [°]. No input is required.
The cutting angle γ is fixed at 90 [°]. No input is required.
| Volume V |
[Source of the Formula] Atsuta Jingu Mathematical Inscription
Copy the URL below, click it online to open the site, and you can find it under “Aichi Prefecture, Atsuta Jingu 4.”
Wasan no Yakata: http://www.wasan.jp/
[Paper Proving the Correctness of the Formula] Formula for the Volume of a Truncated Cone and Its Derivation and Proof
Copy the URL below and click it online to view the paper.
https://drive.google.com/file/d/1AuNbqzPdPuiJhn0739EdncBrsuF9Ik-o/view?usp=sharing
[Related Programs] List of the “Volume Calculation for Truncated Cones” Series
If the conditions cannot be calculated using the program above, you can select one of the following options to perform the calculation.
(1) Volume Calculation for Truncated Cones 1. Vertical Truncation°
(2) Volume Calculation for a Truncated Cone 2. Oblique Cut Where the Cut Surface Touches the Base°
(3) Volume Calculation for a Truncated Cone 3A. Oblique Cut Angle Smaller Than the Base Angle°
(4) Volume Calculation for a Truncated Cone 3B. Oblique Cut Angle = Base Angle °
(5) Volume Calculation for a Truncated Cone 3C. Oblique Cut Angle Greater Than the Base Angle °
[Other Equivalent Programs] The same calculations as this program can be performed on the high-precision calculation website at
https://keisan.site/exec/system/14748552221122 “Mathematics/Physics: Volume/Surface Area ⇒ Volume of a Straight Cone with a Portion Removed.”At first glance, the formula presented there may appear completely different from the one introduced in this program, but upon further manipulation, it becomes clear that they are actually identical. However, because that site uses the notation “inverse hyperbolic functions,” there is a limitation: if the cut-off portion exceeds half of the cone, an error occurs and the calculation cannot be performed.Since this program does not have that constraint, it offers greater flexibility: you can cut the cone at any point between 0 and π.
Copy the URL below, click it online to open the site, and you can find it under “Aichi Prefecture, Atsuta Jingu 4.”
Wasan no Yakata: http://www.wasan.jp/
[Paper Proving the Correctness of the Formula] Formula for the Volume of a Truncated Cone and Its Derivation and Proof
Copy the URL below and click it online to view the paper.
https://drive.google.com/file/d/1AuNbqzPdPuiJhn0739EdncBrsuF9Ik-o/view?usp=sharing
[Related Programs] List of the “Volume Calculation for Truncated Cones” Series
If the conditions cannot be calculated using the program above, you can select one of the following options to perform the calculation.
(1) Volume Calculation for Truncated Cones 1. Vertical Truncation°
(2) Volume Calculation for a Truncated Cone 2. Oblique Cut Where the Cut Surface Touches the Base°
(3) Volume Calculation for a Truncated Cone 3A. Oblique Cut Angle Smaller Than the Base Angle°
(4) Volume Calculation for a Truncated Cone 3B. Oblique Cut Angle = Base Angle °
(5) Volume Calculation for a Truncated Cone 3C. Oblique Cut Angle Greater Than the Base Angle °
[Other Equivalent Programs] The same calculations as this program can be performed on the high-precision calculation website at
https://keisan.site/exec/system/14748552221122 “Mathematics/Physics: Volume/Surface Area ⇒ Volume of a Straight Cone with a Portion Removed.”At first glance, the formula presented there may appear completely different from the one introduced in this program, but upon further manipulation, it becomes clear that they are actually identical. However, because that site uses the notation “inverse hyperbolic functions,” there is a limitation: if the cut-off portion exceeds half of the cone, an error occurs and the calculation cannot be performed.Since this program does not have that constraint, it offers greater flexibility: you can cut the cone at any point between 0 and π.
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