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Whether you’re tackling a geometry homework problem, designing curved surfaces in engineering, or simply curious about 3D shapes, knowing how to calculate the surface area of a spherical cap is incredibly useful. You’ll find this concept everywhere—from space exploration to biomedical devices. While our Cap Surface Area Calculator makes the math effortless, this guide explores the geometry, practical uses, and even some fascinating stories behind this unique shape.
Want to learn more about other shapes? Try our full collection of Surface Area Calculators.
What Exactly Is a Spherical Cap?
A spherical cap is formed when a flat plane slices through a sphere, leaving behind a dome-like, curved surface. Think of cutting off the top of an orange—that curved piece you’re left with is a classic example of a spherical cap. It’s not quite a hemisphere, just a segment of the sphere’s outer shell.
Three main dimensions define a spherical cap:
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The radius of the original sphere (R)
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The height of the cap (h), measured from the flat base to the top of the curve
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The radius of the circular base created by the cut
What makes a spherical cap unique is how it curves in 3D space. Its surface isn’t flat like a circle or simple like a cylinder—it bends, and this curvature is exactly what the formula accounts for.
Be careful not to confuse spherical caps with similar terms like:
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Hemisphere (half a sphere)
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Spherical segment (a band cut by two parallel planes)
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Domes, which are architectural structures inspired by the shape, but not strictly defined mathematically
Try out Surface Area of Cylinder Calculator
The Formula Behind Cap Surface Area
Although it may seem complex, the surface area formula for a spherical cap is refreshingly simple:
A=2πrh
Where:
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Ais the surface area of the cap, -
ris the radius of the original sphere, -
his the height of the cap (from the flat base up to the curved surface).
This formula works because it captures how a cap is just a curved slice of the sphere’s surface. It takes both the curvature (from the radius) and the vertical depth (from the height) into account.
For those who enjoy calculus, this formula is derived using solids of revolution, where a circular arc is rotated around an axis to form the cap and its surface is calculated by integrating small bands.
Just remember to keep your units consistent: if R is in centimeters, h must be too. Your final answer will be in square units like cm² or m².
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How to Use This Tool
Whether you're working on a geometry assignment or designing a technical model, the Cap Surface Area Calculator helps you quickly and accurately find the curved surface area of a spherical cap. Here’s how to use it in a few simple steps:
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Enter the sphere’s radius (R) – the distance from its center to any point on the surface
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Input the height of the cap (h) – from the base up to the curved surface
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Click "Calculate"
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Instant Result – get the curved surface area instantly in your chosen units
Whether you're modeling a component for 3D printing, working on a physics project, or solving math problems, this tool saves time and ensures accuracy.
Archimedes: The Genius Behind Spherical Geometry
Over 2,000 years ago, Archimedes explored shapes that still challenge and inspire us today. While he’s known for his discoveries in buoyancy and levers, he was also one of the first to explore curved 3D surfaces—including spheres and their sections.
He discovered that the surface area of a sphere is exactly four times that of its largest circle (great circle). That insight laid the foundation for how we understand spherical caps today. Archimedes was so proud of this discovery that he asked for a sphere and a cylinder to be engraved on his tombstone—a tribute to the beauty of mathematical geometry.
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Try out Surface Area of Elipsoid Calculator
Related Concepts to Know
1. Total Surface Area of a Sphere
Before diving into spherical caps, remember that the surface area of a complete sphere is given by:
A=4πr2
where:
r is the radius of the sphere. This represents the full, outer surface.
2. Surface Area of a Spherical Cap
A cap is only a portion of a sphere, so its surface area is smaller. The formula is:
A=2πRh
Where:
h is the height of the cap (from the flat base to the curved top), and R is the radius of the original sphere.
3. Volume of a Spherical Cap
If you need to find how much space the cap takes up, use the formula:
V=1/3πh²(3R−h)
This comes in handy for modeling liquids, domes, or biological structures.
Check out Math section to solve math quickly and easily
