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Spherical cap volume calculator

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What is a spherical cap?

A spherical cap is a three-dimensional geometric figure that results when a sphere is sliced by a plane. The cap is the smaller section of the sphere that remains. This shape is often visualized as the “cap” on a sphere, similar to how the lens of a contact lens fits on an eye.

Key components

  • Radius of the sphere (r): The distance from the center of the sphere to its surface.
  • Height of the cap (h): The distance from the base of the cap to the highest point on the cap.
  • Radius of the base of the cap (a): The radius of the circle that forms the base of the cap.

Formula for spherical cap volume

The volume VV of a spherical cap can be calculated using the following formula:

V=(π×h23)×(3r−h)V = \left( \frac{\pi \times h^2}{3} \right) \times (3r - h)

Alternatively, the volume can also be calculated by:

V=16×π×h×(3a2+h2)V = \frac{1}{6} \times \pi \times h \times (3a^2 + h^2)

These formulas are derived from the integration of the volume element of a sphere over the desired limits. In these equations:

  • rr is the sphere’s radius,
  • hh is the cap’s height,
  • aa is the radius of the base of the cap.

Derivation of the formula

The derivation of the spherical cap volume formula involves integrative calculus, which allows us to compute the sum of infinitesimal circular slices comprising the cap. By integrating these slices along the height of the cap, the volume is determined.

Practical applications

  1. Engineering: Spherical caps can model domes, tanks, and other structural elements.
  2. Astronomy: Used in analyzing celestial bodies and their interactions.
  3. Manufacturing: Utilized in fabricating lenses and other curved surfaces.

Examples

Let’s illustrate how to use these formulas with examples.

Example 1: Calculating volume with sphere radius and cap height

Suppose we have a sphere with a radius r=10r = 10 cm, and we are measuring a cap with a height of h=3h = 3 cm.

Using the formula:

V=(π×323)×(3×10−3)V = \left( \frac{\pi \times 3^2}{3} \right) \times (3 \times 10 - 3) V=(π×93)×27V = \left( \frac{\pi \times 9}{3} \right) \times 27 V=3π×27=81π≈254.47 cm3V = 3\pi \times 27 = 81\pi \approx 254.47 \text{ cm}^3

Example 2: Calculating volume with cap base radius

For a cap where the base radius a=8a = 8 cm, and cap height h=5h = 5 cm:

V=16×π×5×(3×82+52)V = \frac{1}{6} \times \pi \times 5 \times (3 \times 8^2 + 5^2) V=16×π×5×(3×64+25)V = \frac{1}{6} \times \pi \times 5 \times (3 \times 64 + 25) V=16×π×5×217≈568.1 cm3V = \frac{1}{6} \times \pi \times 5 \times 217 \approx 568.1 \text{ cm}^3

Notes

  • Ensure that all measurements are in consistent units before calculating.
  • Use precise values for π\pi (such as 3.14159) for more accurate results.

Frequently asked questions

How to calculate the volume of a spherical cap?

Use the provided formulas, depending on whether you have the height of the cap and the sphere’s radius or the radius of the base of the cap.

What is the difference between a sphere and a spherical cap?

A sphere is a complete three-dimensional shape, whereas a spherical cap is a portion of the sphere defined by slicing with a plane.

What happens if the plane passes through the sphere’s center?

The resulting shape would no longer be a cap but a hemisphere if divided into two equal parts. The volume of a hemisphere can be calculated using the formula: V=23πr3V = \frac{2}{3} \pi r^3 or use our hemisphere volume calculator.

Can the volume formula be used for hemispheres?

No, the spherical cap volume formula is specific to caps. Hemispheres have their specific volume formula.

How many degrees of a sphere is a spherical cap?

A spherical cap’s angle depends on its height and is computed separately using the central angle in spherical coordinates.

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