Spheres
A sphere is every point in space at a fixed distance from the center.
About this topic
A sphere is the set of all points in space at a fixed distance from a centre. Its surface area is 4πr² and its volume is (4/3)πr³. Archimedes discovered that a sphere's volume is two-thirds that of the cylinder that tightly encloses it.
The sphere has the smallest surface area for a given volume, which is why bubbles and drops are round. Planets, balls and cell models are all described by these formulas, and doubling the radius multiplies volume by eight.
Worked example: a ball has radius 6 cm. Its surface area is 4π(36) = 144π ≈ 452 square cm, and its volume is (4/3)π(216) = 288π ≈ 905 cubic cm. If the radius doubles to 12, the surface area is four times larger and the volume eight times larger. A hemisphere has half the volume plus a flat circular face when finding total surface.
Basketballs, planets and bubbles are spheres, and storage tanks use spherical shapes to withstand pressure. Typical errors include using the diameter as the radius, forgetting to cube r in the volume, and writing 4/3 πr² by mixing the formulas. To check, the surface area is exactly four times the area of a great circle. Estimate your answers: a sphere fits inside a cube of side 2r, so its volume must be less than 8r³.
Key ideas
- V = 4/3 πr³
- Surface area = 4πr²
- Great circles split a sphere in half
Quick check
Use cases: where Spheres helps
- Homework: review the key idea behind Spheres before you start a problem set.
- Exam revision: use the quick check to test whether you can recall Spheres without notes.
- Visual learning: play with the interactive lab to see how Spheres changes when you adjust its inputs.
- Teaching a friend: explain Spheres in your own words, then compare your version with the key ideas.
Number and algebra skills are used in science, engineering, budgeting and computing.
Frequently asked questions
What is Spheres?
A sphere is every point in space at a fixed distance from the center.
What will I learn in Spheres?
V = 4/3 πr³. Surface area = 4πr². Great circles split a sphere in half.
Is Spheres part of Mathematics?
Yes. It is one of 200 Mathematics topics on LearnViz HighSchoolEdu, with 3 key ideas, a live animation and a quick check.
Does the Spheres lab work on a phone or tablet?
Yes. The animation resizes to fit your screen and supports touch, so you can drag, tap and slide with your finger.
How should I study Spheres?
Read the summary and key ideas first, play with the interactive lab to see the idea in motion, then answer the quick check. If you get a question wrong, go back to the key idea it tests.
Do I need an account, and is my progress saved?
No account is needed. Marking a topic as learned is saved in your own browser, so it stays on that device and is not sent to a server.