Sphere Calculator
Calculate sphere properties including volume, surface area, radius, diameter, and circumference. Enter any value and get all other measurements instantly.
V = (4/3)pir^3 | A = 4pir^2Sphere Properties
Volume
523.5988 cm^3
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All Sphere Properties
Radius (r)
5 cm
Diameter (d)
10 cm
d = 2r
Circumference (C)
31.4159 cm
C = 2pir
Volume (V)
523.5988 cm^3
V = (4/3)pir^3
Surface Area (A)
314.1593 cm^2
A = 4pir^2
Exact Values (in terms of pi)
Volume
166.6667pi cm^3
Surface Area
100pi cm^2
Circumference
10pi cm
Sphere Formulas
Basic Formulas
- Volume: V = (4/3)pir^3
- Surface Area: A = 4pir^2
- Circumference: C = 2pir
- Diameter: d = 2r
Derived Formulas
- r from V: r = (3V / 4pi)^(1/3)
- r from A: r = sqrt(A / 4pi)
- r from C: r = C / 2pi
- r from d: r = d / 2
Fun Fact: Archimedes proved that the volume of a sphere is exactly 2/3 of the cylinder that contains it (same height and diameter). He considered this his greatest discovery!
Sphere Properties
Volume
523.5988 cm^3
?How to Calculate Sphere Properties
Sphere formulas: Volume = (4/3) x pi x r^3 (four-thirds pi r cubed). Surface Area = 4 x pi x r^2 (four pi r squared). Circumference = 2 x pi x r. Diameter = 2 x radius. From any measurement, you can calculate all others. A sphere is a perfectly round 3D shape where every point on the surface is equidistant from the center.
What is a Sphere?
A sphere is a perfectly round three-dimensional geometric shape where every point on its surface is exactly the same distance (radius) from the center. Unlike a circle (which is 2D), a sphere exists in 3D space. Common examples include balls, planets, bubbles, and oranges. The sphere has the remarkable property of enclosing the maximum volume with the minimum surface area of any shape.
Key Facts About Spheres
- Volume = (4/3) x pi x r^3 (approximately 4.189 x r^3)
- Surface Area = 4 x pi x r^2 (approximately 12.566 x r^2)
- Circumference = 2 x pi x r (same as a circle with same radius)
- Diameter = 2 x radius; radius = diameter / 2
- A sphere has the smallest surface area for a given volume of any shape
- The volume of a sphere is exactly 2/3 of the cylinder that contains it
- Spherical cap volume = (pi x h^2 x (3r - h)) / 3
- All points on a sphere surface are equidistant from the center
Practice Sphere Problems
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Practice with 4 problems to test your understanding.
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Quick Answer
Sphere formulas: Volume = (4/3) x pi x r^3 (four-thirds pi r cubed). Surface Area = 4 x pi x r^2 (four pi r squared). Circumference = 2 x pi x r. Diameter = 2 x radius. From any measurement, you can calculate all others. A sphere is a perfectly round 3D shape where every point on the surface is equidistant from the center.
Frequently Asked Questions
The volume of a sphere is V = (4/3)pir^3, where r is the radius. This formula can be derived using calculus by integrating circular cross-sections. For a sphere with radius 5, the volume is (4/3) x 3.14159 x 125 = 523.6 cubic units.
The surface area of a sphere is A = 4pir^2, where r is the radius. Interestingly, this is exactly 4 times the area of a circle with the same radius. This formula shows that surface area grows with the square of the radius.
To find the radius from volume, rearrange the volume formula: r = cube root of (3V / 4pi). For example, if the volume is 523.6 cubic units, r = cube root of (3 x 523.6 / 4 x 3.14159) = cube root of 125 = 5.
A spherical cap is the portion of a sphere cut off by a plane. If you slice off the top of a sphere, the dome-shaped piece is a spherical cap. Its volume is V = (pih^2(3r - h))/3, where h is the height of the cap and r is the sphere radius.
A hollow sphere (or spherical shell) has an outer radius and an inner radius, like a tennis ball or Earth's crust. Its volume is the difference between the outer and inner sphere volumes: V = (4/3)pi(R^3 - r^3), where R is outer radius and r is inner radius.
A sphere encloses the maximum volume for a given surface area (or equivalently, has minimum surface area for a given volume). This is why bubbles form spheres - nature minimizes surface tension. This property is known as the isoperimetric inequality.
A sphere inscribed in a cylinder (where the cylinder height equals the sphere diameter) has exactly 2/3 the volume of the cylinder. This was discovered by Archimedes and he was so proud of it that he requested it be engraved on his tombstone.
Sphere packing is the arrangement of non-overlapping spheres in a container. The densest packing is face-centered cubic or hexagonal close packing, achieving about 74% efficiency (26% is empty space). Random packing achieves about 64%.
Last updated: 2025-01-15
Sphere Properties
Volume
523.5988 cm^3