CalcKit
Spherical Cap Calculator
Table of Contents
- Spherical cap definition
- Parts of a spherical cap
- Spherical cap, segment and sector
- Frequently asked questions
Spherical cap definition
A spherical cap is the part of a sphere that is cut off by a plane. It has a flat circular base and a curved, dome-shaped surface, which is why it is also called a spherical dome.
If the cutting plane passes through the centre of the sphere, the cap is exactly half the sphere, a hemisphere. A cap smaller than a hemisphere is flatter, and a cap larger than a hemisphere is more than half of the ball.
Spherical caps are everywhere: domes on buildings, contact lenses, the lid of a bowl, a slice cut from a ball, and even the shape of a drop of water resting on a surface.
Parts of a spherical cap
The seven parts of a spherical cap mentioned in this tool are its height (h), the radius of its base (r), the radius of the sphere (R), base area (Ab), curved cap area (Ac), total area (A) and volume (V).
Height of spherical cap (h)
The height of a spherical cap is the distance from the centre of its flat base to the top of its dome.
It can be found with the following formulas:
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If you know the cap area and the radius of the sphere:
h = Ac / (2 * π * R) -
If you know the cap area and the base radius:
h = √(Ac / π - r²) -
If you know the total area and the base radius:
h = √(A / π - 2 * r²)
Radius of spherical cap (r)
The radius of a spherical cap is the radius of its flat circular base.
It can be found with the following formulas:
-
If you know the radius of the sphere and the height:
r = √(2 * R * h - h²) -
If you know the base area:
r = √(Ab / π) -
If you know the cap area and the height:
r = √(Ac / π - h²) -
If you know the total area and the height:
r = √((A / π - h²) / 2) -
If you know the volume and the height:
r = √((6 * V / (π * h) - h²) / 3)
Radius of sphere (R)
The radius of the sphere is the radius of the whole ball the cap was cut from.
If you know the base radius and the height of the cap:
R = (r² + h²) / (2 * h)
Base area (Ab)
The base area of a spherical cap is the area of its flat circular base.
It can be found with the following formulas:
-
If you know the base radius:
Ab = π * r² -
If you know the total area and the cap area:
Ab = A - Ac
Area of spherical cap (Ac)
The area of a spherical cap is the area of its curved, dome-shaped surface only, without the base.
It can be found with the following formulas:
-
If you know the base radius and the height:
Ac = π * (h² + r²) -
If you know the radius of the sphere and the height:
Ac = 2 * π * R * h -
If you know the total area and the base area:
Ac = A - Ab
Total area (A)
The total area of a spherical cap is the area of its curved surface and its flat base together:
A = Ab + Ac
Volume (V)
The volume of a spherical cap is the amount of space inside it, between the dome and the base.
If you know the base radius and the height:
V = π * h * ((3 * r²) + h²) / 6
Spherical cap, segment and sector
Three shapes are cut from a sphere, and they are easy to mix up:
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Spherical cap: the part of a sphere cut off by one plane. It has one flat base and a curved top.
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Spherical segment: the part of a sphere between two parallel planes, like a slice of a ball. It has two flat bases. A spherical cap is a segment whose top base has shrunk to a point.
-
Spherical sector: a spherical cap together with the cone that joins its base to the centre of the sphere, like a scoop of ice cream on a cone.
Frequently asked questions
How do you find the volume of a spherical cap?
Use V = π * h * ((3 * r²) + h²) / 6, where h is the height of the cap and r the radius of its base. For a cap with r = 4 and h = 2, that is π × 2 × (3 × 4² + 2²) / 6 ≈ 54.45 cubic units.
How do you find the radius of the sphere from a spherical cap?
Measure the base radius r and the height h of the cap, then use R = (r² + h²) / (2 * h). This is how the size of a dome or a lens is worked out from its width and depth.
What is the surface area of a spherical cap?
The curved surface has an area of Ac = 2 * π * R * h, which is the same as π * (h² + r²). Adding the flat base, π * r², gives the total area.
Is a hemisphere a spherical cap?
Yes. A hemisphere is a spherical cap whose height equals the radius of the sphere, cut by a plane through the centre.