A spherical zone is a three-dimensional geometric shape formed by slicing a sphere with two parallel planes. The region of the sphere that lies between these two cuts is what we call the spherical zone. Unlike a spherical cap, which is formed by making a single cut and removing the top or bottom portion of the sphere, a spherical zone retains a band or slice of the sphere’s surface between two planes.
The spherical zone has two circular faces — the upper face with radius b and the lower face with radius a. These circles are typically not the same size unless the cuts are made symmetrically. The height of the spherical zone, labeled h, is the vertical distance between the two planes that slice the sphere.
This geometric form appears in a wide range of practical applications. It can be found in architectural domes, pressure vessels, and segments of lenses or optical components. It is also commonly used in mathematical modeling and physics to calculate volumes and surface areas of partial spheres.
In addition to the circular faces, the curved surface of the spherical zone forms part of the sphere’s original outer shell. This surface area is known as the lateral surface area (A). When you add the areas of the two circular faces to the lateral surface area, you get the total surface area (AT) of the zone.
Using our spherical zone calculator, you can enter specific values such as the radii a and b, the height h, or the radius of the original sphere r. Based on these inputs, the calculator provides you with precise results for:
Whether you're a student, engineer, or simply curious about geometric shapes, understanding the spherical zone can help you visualize and calculate parts of a sphere that are often encountered in real-world designs and natural forms.
a, b, r, and h?To calculate the volume and surface area of a spherical zone, you need to provide certain values that describe its shape. Here's what each variable represents:
a – Radius of the lower circular face of the spherical zone.b – Radius of the upper circular face of the spherical zone.r – Radius of the original sphere from which the zone is sliced. This helps determine the curvature of the zone.h – Vertical height of the zone (the distance between the two circular faces).The calculator is designed to be flexible. You don’t need to enter all four values. Instead, you can calculate missing values based on the combination of inputs you provide:
a, b, r, and h. The calculator can then compute the missing one.
a, b, and r, the height h can be calculated automatically.
a, b, and h, the calculator can determine the radius of the sphere r.
This flexibility makes it easy to work with the measurements you already have and still get accurate results for volume and surface area.
Using the Spherical Zone Area and Volume Calculator is simple and intuitive. Whether you're working on a geometry assignment, engineering task, or just exploring 3D shapes, this tool helps you quickly calculate volume and surface areas of a spherical zone based on the measurements you provide.
a – Radius of the lower circular faceb – Radius of the upper circular facer – Radius of the original sphereh – Height of the zone (distance between the two circular faces)r or h, depending on what information is available.
h), you can leave it empty. The calculator will compute it based on the other values.
V – Volume of the spherical zoneA – Lateral surface areaAT – Total surface area (including both circular faces)h – Height (if it was not provided and calculated instead)The calculator is ideal for both quick estimates and detailed work, and it supports various use cases depending on which values you start with. It’s designed to be as helpful as possible even when you only have partial data.
After entering your values and clicking “Calculate,” the spherical zone calculator will display several key results that describe the shape and size of the zone. Here’s what each result means and how it relates to the geometry of the spherical zone:
V: Volume
This is the amount of three-dimensional space occupied by the spherical zone. It is measured in cubic units (such as cm³, m³, etc.) and is calculated based on the radii of the two circular faces (a and b) and the height h of the zone. The volume gives you an idea of how much material or space is enclosed within the shape.
A: Lateral Surface AreaThis represents the curved surface area of the zone — the outer shell that wraps around between the two circular faces. It does not include the top and bottom surfaces. This is especially useful if you are painting, coating, or wrapping the zone and need to know the side surface only.
AT: Total Surface Area
The total surface area includes the lateral surface area (A) plus the areas of both the upper and lower circular faces. This gives you the complete surface coverage of the entire shape. It's useful when you need to calculate how much material is needed to cover the entire zone.
h: Height
If you didn’t provide the height h as an input, the calculator will compute it for you. This value represents the vertical distance between the two parallel cuts that form the top and bottom of the spherical zone. It’s a key measurement in understanding the thickness or depth of the zone.
These results help you fully understand the dimensions and properties of the spherical zone, allowing for accurate measurements and applications in construction, science, and design.
The spherical zone calculator uses well-established mathematical formulas to compute the volume, surface areas, and height of the zone based on the values you provide. Here's a breakdown of each formula used in the background:
V)The volume of a spherical zone is calculated using the following formula:
V = (π · h / 6) · (3a² + 3b² + h²)
This formula combines the height of the zone and the radii of the two circular faces to determine the total space inside the shape.
A)The curved or lateral surface area is given by:
A = 2π · r · h
Here, r is the radius of the original sphere, and h is the height of the zone. This gives the surface area of the curved side between the two circular faces.
AT)The total surface area includes both the lateral surface and the two circular faces:
AT = π(2r · h + a² + b²)
This formula adds the areas of the two ends to the curved side area, giving you the complete surface coverage.
h)If the height is unknown, it can be calculated from the radii of the circular faces and the radius of the sphere:
h = √(r² − b²) − √(r² − a²)
This comes from the geometry of the sphere and the vertical positions of the slicing planes.
r)
If r is not known, it can be found using:
r = √[a² + ((a² − b² − h²) / 2h)²]
This allows the calculator to work even when only a, b, and h are given.
These formulas are built into the calculator to ensure accurate and reliable results, whether you’re working with known or partial inputs.
Let’s walk through an example to see how the spherical zone calculator works in practice. Suppose you have a spherical zone with the following measurements:
You did not provide the height h, so the calculator will compute it automatically using the formula:
h = √(r² − b²) − √(r² − a²) = √(6² − 2²) − √(6² − 4²) = √(36 − 4) − √(36 − 16) = √32 − √20 ≈ 5.657 − 4.472 ≈ 1.185 units
V)V = (π · h / 6) · (3a² + 3b² + h²) = (π · 1.185 / 6) · (3·4² + 3·2² + 1.185²) = (π · 1.185 / 6) · (48 + 12 + 1.404) = (π · 1.185 / 6) · 61.404 ≈ 0.621 · 61.404 ≈ 38.15 units³
A)A = 2π · r · h = 2π · 6 · 1.185 ≈ 44.65 units²
AT)AT = π(2r · h + a² + b²) = π(2·6·1.185 + 16 + 4) = π(14.22 + 20) = π · 34.22 ≈ 107.49 units²
So based on the input values, the calculator would display the following results:
h): ≈ 1.185 unitsV): ≈ 38.15 units³A): ≈ 44.65 units²AT): ≈ 107.49 units²This example shows how easily you can use the calculator to understand the geometry of a spherical zone using just a few inputs.
A spherical cap is formed by slicing a sphere with a single plane, creating a dome-like shape. A spherical zone, on the other hand, is formed by slicing the sphere with two parallel planes, leaving a band-like shape between the two cuts.
No. The calculator requires at least three values among a, b, r, and h. With three values, it can calculate the fourth and then proceed to compute volume and surface areas.
You can use any consistent unit (e.g., centimeters, inches, meters), but all input values must be in the same unit. The results will automatically be in cubic units for volume and square units for surface areas.
The geometry of a spherical zone depends heavily on how the values relate to one another. If you change which value is missing or adjust your inputs, it affects the overall shape, leading to different results. Always double-check your known values.
Absolutely! Spherical zones appear in many fields such as architecture (domes), mechanical design (tanks, lenses), and science. This calculator helps professionals and students estimate material quantities, surface coatings, and more.
If the combination of values doesn’t make geometric sense (for example, a and b are too large compared to r), the calculator may display an error or unexpected result. Always ensure your inputs are realistic and physically possible.