Sphere Calculator
Calculate sphere volume, surface area, diameter, and great circle circumference from radius with step-by-step mathematical deduction.
Did this calculation save you time?
Student ProjectHi! I'm a student developer building Calculat in my spare time. I was tired of searching for basic math tools and having to click through 10 spammy popups, loan ads, and cookie trackers.
I keep this website 100% free, private, and ad-free. If this helped you with your homework, project, or finances today, bookmarking this page or telling a friend helps me keep building more free tools!
What Is a Sphere Calculator?
A sphere is a perfectly round three-dimensional geometrical object consisting of all points in space that are at an equal distance (radius r) from a central point.
How to Use This Calculator
- Enter the radius (r) of the sphere.
- View the calculated Volume, Surface Area, Diameter, and Great Circle Circumference.
- Follow the step-by-step algebraic steps.
Sphere Volume and Surface Area Formulas
V = \frac{4}{3}\pi r^3, \quad A = 4\pi r^2, \quad C = 2\pi r, \quad d = 2rSurface area is exactly four times the area of its great circle (4πr²). Volume is four-thirds pi radius cubed, proven historically by Archimedes.
Worked Example
Scenario: Find volume and surface area of a sphere with radius r = 6 cm
Diameter: d = 2 × 6 = 12 cm
Circumference: C = 2 × π × 6 = 12π ≈ 37.6991 cm
Surface Area: A = 4 × π × 6² = 4 × π × 36 = 144π ≈ 452.3893 cm²
Volume: V = ⁴⁄₃ × π × 6³ = ⁴⁄₃ × π × 216 = 288π ≈ 904.7787 cm³
Tips & Key Notes
- A sphere has the smallest surface area of any geometric shape for a given volume.
- To find radius from volume, isolate r: r = ∛(3V / (4π)).
- To find radius from surface area, isolate r: r = √(A / (4π)).
Frequently Asked Questions
What is a great circle on a sphere?
A great circle is the largest circle that can be drawn on a sphere, formed by the intersection of the sphere with a plane passing through its center.
What is a hemisphere?
A hemisphere is exactly half of a sphere. Its volume is (2/3)πr³ and total surface area (including circular base) is 3πr².
How did Archimedes discover the sphere formula?
Archimedes proved that a sphere inscribed in a cylinder has exactly 2/3 the volume and 2/3 the surface area of the cylinder.
How do you find the diameter of a sphere?
Diameter is simply double the radius: d = 2r.
Related Calculators
Explore similar toolsCircle Calculator
Calculate circle area, circumference, diameter, radius, arc length, and sector area with step-by-step geometric derivations.
Cylinder Calculator
Calculate right circular cylinder volume, total surface area, lateral (curved) surface area, and base area from radius and height.
Cone Calculator
Calculate right circular cone volume, total surface area, lateral (conical) surface area, slant height, and base area with step-by-step steps.