Spheres
Volume and surface area of a sphere.
Key idea
Volume and surface area of a sphere.
In depth
A sphere is all points at distance r from a centre. Its surface area is 4πr², and its volume is (4/3)πr³. Archimedes showed the volume is two thirds of the cylinder that encloses it.
Worked example
r = 3: SA = 36π ≈ 113.1; V = 36π ≈ 113.1 (they happen to be equal for r = 3).
Formula
Real-world uses
- Calculating the volume of a ball, a bubble or a planet
- The amount of material in a spherical tank
Common mistake
Using r² in the volume or r³ in the area.
How to learn this
- Read the key idea and say it in your own words.
- Work through the worked example, then try a similar problem.
- Use the interactive lesson’s visual and mini-quiz to check the pattern.
Frequently asked questions
What is Spheres?
Volume and surface area of a sphere. It is a topic at the advanced level in Geometry.
What is the formula for Spheres?
The key formula is V = 4/3 πr³, SA = 4πr².
Where is Spheres used in real life?
Real-world examples: Calculating the volume of a ball, a bubble or a planet; The amount of material in a spherical tank.
What should I learn before Spheres?
Make sure you are comfortable with Pyramids, the previous topic in Geometry.
What is a common mistake with Spheres?
Using r² in the volume or r³ in the area.
How can I practise Spheres?
Open the interactive lesson, play with the visual, then answer the three-question mini-quiz. Getting two right marks the topic complete.
What comes after Spheres?
Move on to Composite Solids next.
Related advanced topics in Geometry
- Heron’s Formula – Triangle area from three sides.
- Regular Polygon Area – Use apothem and perimeter.
- Arc Length of a Circle – Part of the circumference.
- Sector Area – Part of the circle’s area.
- Central and Inscribed Angles – An inscribed angle is half the central angle on the same arc.
- Chord and Tangent Properties – A tangent is perpendicular to the radius at the point of contact.