Cone Frustum Volume from Angle

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Dec 10, 2024, 2:43:20 PM

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Dec 10, 2024, 2:33:23 PM

`V = 1/3 * pi * h(a^2+a * r+r^2)`

`(alpha)"Side Angle"`

`(r) "bottom radius"`

`(h) "height"`

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b59101ac-b703-11ef-98f1-bc764e203090

The Cone Frustum Volume from Angle calculator computes the volume of a frustum (slice) of a cone based Diagram of cone frustum with side angle on the bottom radius and height in between top and bottom and the angle of the side.

INSTRUCTION: Choose units and enter the following:

(α) Side Angle
(r) Radius of Bottom
(h) Height in between top and bottom

Volume of the Cone Section (V): The calculator computes the volume (V) in cubic meters (m³). However this can be automatically converted to many other volume units (e.g. gallons or cubic feet) via the pull-down menu.

The Math

The frustum of a cone is a section of a cone also known as a truncated cone. The frustum of a cone is created by parallel planes that go through the cone and are normal (perpendicular) to the center axis of the cone. The shape of the top and bottom are perfect circles whose centers are directly aligned. The planes are separated by the height (h). The formula for the Volume of the Frustum of a cone is as follows:

V = ¹/₃ • π • h(a²+a•b+b²)

where:

V is the volume of the frustum (section) of a cone
a is the radius of one end
b is the radius of the other end
h is the height

Cone Calculators

Cone Surface Area based on cone height and cone base radius
Cone Volume based on cone height and cone base radius
Cone Mass or Weight as a function of the volume and mean density
Cone Frustum Surface Area based on the two heights and two radii
Cone Frustum Volume based on the two heights and two radii
Cone Frustum Weight or Mass based on the cone frustum volume and mean density
Cone Shell Volume a.k.a. hollow cone volume
Cone Shell Mass or Weight a.k.a. hollow cone mass
Moment of inertia of a cone shaped object based on mass and rotation axis
Moment of inertia of a cone shaped object based on mass and distance to apex (rotation around apex)
Compute the total surface are of the frustum of a cone: This includes the top and bottom surfaces.
Compute the lateral surface area (using L) of the frustum of a cone.
Cone Frustum Volume from Angle
Cone Frustum Height from Volume and Outer Side Angle: Computes height based on volume, small radius and outer side angle.
Cone Frustum Height from Volume and Inner Side Angle: Computes height based on volume, small radius and inner side angle.

See Also

Frustum Calculator - Equations for volume, mass and surface areas for various frustums.
Planter Calculator - Equations for home and garden planters.
Volume Calculator - Volume equations for various geometric shapes
Mass Weight Calculator - Equations for the mass and weight of objects based on their mean density and shape

Volume is a three dimensional measurement of the amount of space taken up by an object. Volume units are cubic measurements for solid objects such as cubic inches and cubic meters. Fluids have separate volume units such as liters, fluid ounces, cups, gallons, and barrel.

The volume of an object can measured by the liquid it displaces or be calculated by measuring its dimensions and applying those dimensions to a formula describing its shape. Many such calculations are available in the following list of calculators.

In many cases, the calculators are for a column with a geometric shaped base and vertical sides. One basic formula for volume is area times a Height when the volume has vertical sides.

Volume Calculators

Reference

Equation 4.42 Spiegel, Murray R. "Chapter 4." Mathematical Handbook of Formulas and Tables. New York: McGraw-Hill, 1968. N. pag. Print.

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