Sphere Calculator
Calculate volume, surface area, diameter, and all sphere properties from any known value
Selected Mode
Calculate all sphere properties from any known value Calculate spherical cap surface area, volume, and base radius
The distance from center to surface
The distance across the sphere through center (d = 2r)
Total area covering the sphere (A = 4πr²)
Space enclosed within the sphere (V = 4/3 πr³)
Must be ≤ 2r (diameter)
Sphere Properties
Radius (r)
Diameter (d)
Surface Area (A)
Volume (V)
Great Circle Properties
Great Circle Circumference
Great Circle Area
Formulas Used
Diameter
d = 2r
Surface Area
A = 4πr²
Volume
V = (4/3)πr³
Pi (π)
≈ 3.14159...
Spherical Cap Results
Cap Surface Area
Cap Volume
Base Radius
Base Area
Full Sphere Comparison
Full Surface Area
Full Volume
Spherical Cap Formulas
Cap Surface Area
A = 2πrh
Cap Volume
V = (πh²/3)(3r - h)
Base Radius
a = √(h(2r - h))
Constraint
h ≤ 2r
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About Sphere Calculator
What is a Sphere?
A sphere is a perfectly round three-dimensional geometric shape where every point on its surface is equidistant from its center. This distance is called the radius. Spheres appear everywhere in nature and engineering—from planets and celestial bodies to ball bearings, bubbles, and sports balls.
How to Use This Calculator
Basic Properties Mode
- Select Basic Properties from the mode selector
- Choose which value you know: Radius, Diameter, Surface Area, or Volume
- Enter your known value in the input field
- The calculator instantly computes all other sphere properties
Spherical Cap Mode
- Select Spherical Cap from the mode selector
- Enter the Sphere Radius
- Enter the Cap Height (h ≤ 2r)
- View calculated cap surface area, cap volume, and base radius
Sphere Formulas
Basic Sphere Formulas
| Property | Formula | Description |
|---|---|---|
| Diameter | d = 2r | Diameter is twice the radius |
| Surface Area | A = 4πr² | Total area covering the sphere |
| Volume | V = (4/3)πr³ | Space enclosed within the sphere |
| Great Circle | C = 2πr | Circumference of the largest circle |
Spherical Cap Formulas
| Property | Formula | Description |
|---|---|---|
| Cap Surface Area | A = 2πrh | Curved surface of the cap |
| Cap Volume | V = (πh²/3)(3r - h) | Volume of the cap region |
| Base Radius | a = √(h(2r - h)) | Radius of the circular base |
Understanding Sphere Properties
Radius (r)
The radius is the distance from the center of a sphere to any point on its surface. It's the most fundamental measurement, and all other properties derive from it.
Common uses:
- Determining ball sizes in sports
- Calculating planetary dimensions
- Engineering spherical containers
Surface Area (A)
The surface area represents the total area covering the outside of a sphere. The formula A = 4πr² shows that surface area grows with the square of the radius—double the radius and the surface area quadruples.
Common uses:
- Calculating paint or coating needed
- Determining heat transfer rates
- Sizing material for spherical objects
Volume (V)
The volume represents the amount of space enclosed within the sphere. The formula V = (4/3)πr³ shows that volume grows with the cube of the radius—double the radius and the volume increases 8 times.
Common uses:
- Calculating tank capacities
- Determining displacement in physics
- Computing mass from density
Real-World Applications
Engineering & Manufacturing
- Designing pressure vessels and tanks
- Ball bearing specifications
- Spherical joint calculations
Science & Astronomy
- Planetary volume and surface calculations
- Bubble physics and surface tension
- Cell and molecular biology
Everyday Life
- Sports ball sizing
- Decorative sphere dimensions
- Container capacity estimates
Frequently Asked Questions
How do I find the radius from the volume?
Rearrange the volume formula V = (4/3)πr³ to solve for r:
r = ∛(3V / 4π)
How do I find the radius from the surface area?
Rearrange the surface area formula A = 4πr² to solve for r:
r = √(A / 4π)
What is a spherical cap?
A spherical cap is the region of a sphere cut off by a plane. Imagine slicing the top off an orange—that dome-shaped piece is a spherical cap.
How accurate is this calculator?
This calculator uses JavaScript's built-in Math library, providing approximately 15-17 significant digits of precision for π and mathematical operations.