What This Calculator Does
This tool computes the surface area of a sphere from a single input: its radius. A sphere is the three-dimensional counterpart of a circle, and its surface area is exactly four times the area of a great circle (\(\pi r^2\)) that passes through its center. As a bonus, the calculator also reports the sphere's diameter and volume so you have a complete geometric picture from one measurement.
How to Use It
Enter the radius (r) in any unit you like — centimeters, inches, meters, etc. Press calculate and you'll get the surface area in square units of the same measurement system, plus the diameter and volume. If you only know the diameter, divide it by two first to get the radius.
The Formula Explained
The surface area of a sphere is given by:
$$SA = 4\pi r^2$$
Here \(\pi\) (pi) \(\approx 3.14159\), and \(r\) is the radius. The factor of 4 reflects how a sphere's surface "wraps" four flat disks of the same radius. The related volume formula is \(V = \frac{4}{3}\pi r^3\).
Worked Example
Suppose a ball has a radius of 5 cm. Then:
$$SA = 4 \times \pi \times 5^2 = 4 \times \pi \times 25 = 100\pi \approx 314.16 \text{ cm}^2.$$
Its diameter is 10 cm and its volume is \(\frac{4}{3}\pi(125) \approx 523.60 \text{ cm}^3\).
FAQ
Why is sphere surface area \(4\pi r^2\)? It can be derived using calculus (integration of surface elements) and was first proven by Archimedes, who showed a sphere's surface equals the lateral surface of its circumscribing cylinder.
What units does the answer use? Surface area comes out in square units of whatever unit you used for the radius — if \(r\) is in meters, the area is in square meters.
How do I find the radius from surface area? Rearrange the formula: \(r = \sqrt{\frac{SA}{4\pi}}\).
