Circular cross-section of a sphere
A plane at distance d from the centre cuts a circle of radius √(R² − d²).
Key formulas
Right triangle
From the centre to a point on the cut circle: radius R. The perpendicular to the plane has length d. The remaining leg is the circle’s radius ρ.
Example
R=10, d=6 → ρ=8. The section is a circle of radius 8, not 10.
Extreme cases
d=0 → great circle, ρ=R. d=R → a single point.
Worked example
Worked example
Problem. A sphere of radius 10 cm is cut by a plane 6 cm from the centre. Find the radius of the circular cross-section.
Solution. ρ = √(R² − d²) = √(100 − 36) = √64 = 8 cm.
Great circle
Any plane through the centre gives the largest possible section.
Review quiz
Sphere R=13, plane 5 from centre. Section radius is:
Practice questions
- R=13, d=5. Find ρ.
- What is ρ when the plane is 1 cm from the centre of a sphere of radius 5?
Study notes
Is every section a circle?
Yes for a sphere cut by a plane; ellipses appear for other solids.
More in Spatial Geometry
- Pyramid volume: identify the base and perpendicular height
- Distance from a point to a plane using volume
- Finding the angle between a line and a plane
- Writing the equation of a plane in 3D
- Cylinder volume and curved surface area
- Cone volume when slant height is given
- Prism volume from a cross-section
- Sphere volume without the fear of π