Spherical Cap Volume Formula

A spherical cap is a three-dimensional portion of a complete sphere having a circular disc-like base.

• It is also known as a Spherical dome.
• It is formed when a sphere is cut along a plane.
• The height of the spherical cap is the distance between the center of the circular disc of the cap and the highest point on the spherical cap.
• The volume of the spherical cap is obtained by knowing the values of the radius of the sphere and the height of the spherical cap.

Key Terms: Sphere, Volume of a Sphere, Radius, Hemisphere, Circle, Maximum height

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Spherical Cap

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When a sphere or ball is cut off by a plane, then a portion of the sphere is known as a Spherical cap or Spherical dome.

• A spherical cap contains a circular base and a curved surface.
• The base of a spherical dome is formed by cutting a complete sphere along any plane.
• If the plane passes through the center of the sphere, then the height of the spherical cap is equal to the radius of the sphere.
• This type of spherical cap is known as Hemisphere.

Spherical cap obtained from a sphere

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Spherical Cap Volume Formulas

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The volume of a spherical cap is given by

• In terms of the radius of the sphere and the height of the spherical cap

V = \(\frac{\pi h^2}{3}\) (3r – h)

Where

• h = height of the spherical cap
• r = radius of the sphere
• In terms of the radius of the circular base of the cap and height

V = \(\frac{1}{6}\) πh (3a² + h²)

Where a is the radius of the circular base of the spherical cap.

• In terms of the radius of the sphere and polar angle

V = \(\frac{\pi h^3}{3}\) (2 + cosθ)(1 – cos)²

Where θ is known as polar angle i.e. the angle between the lines from the center of the sphere to the maximum height and edge of the spherical cap.

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Solved Examples

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Ques. Find the volume of the spherical cap if the radius of the sphere is 5.5 cm and the height of the spherical cap is 9.6 cm.

Ans. Given

• Radius of the sphere, r = 5.5 cm
• Height of the cap, h = 9.6 cm

Volume of the spherical cap in terms of the radius of the sphere and the height of the cap is given by

V = \(\frac{\pi h^2}{3}\)(3r – h)

On substituting the values, we get

V = \(\frac{3.14 \times 9.6^2}{3}\) (3 x 5.5 – 9.6)

⇒ V = 96.46 x 6.9 = 665.6 cm³

Ques. The height of the circular base of a spherical cap is 3 m. Find the radius of the sphere from which it is formed if the volume of a spherical cap is 165 m³.

Ans. Given

• Volume of the spherical cap, V = 165 m³
• Height of the cap, h = 3 m

Let r be the radius of the sphere, the volume of the spherical cap in terms of the radius of the sphere, and the height of the cap is given by

V = \(\frac{\pi h^2}{3}\)(3r – h)

On substituting the values, we get

165 = \(\frac{3.14 \times 3^2}{3}\) (3r – 3)

⇒ 165 = 9.42 (3r - 3)

⇒ 125 = 28.26r - 28.26

⇒ r = 5.4 m

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Things to Remember

• A part of the sphere is obtained by cutting the sphere along any plane is called spherical Cap.
• A spherical cap contained a curved surface and a circular base.
• The volume of the spherical cap can be calculated by knowing the values of the height of the cap and the radius of the sphere.
• Volume of the spherical cap can also be obtained in terms of the height and radius of the cap.