A sphere is the set of all points in space that are the same distance from one fixed point called the center. Spheres appear in planets, balls, bubbles, and many geometric models, so understanding their parts helps connect geometry to real objects. The most important measurements are radius, diameter, surface area, and volume.
Great circles and small circles describe how flat slices cut through a sphere.
Key Facts
- All radii of the same sphere are congruent.
- Diameter = 2r
- Surface area of a sphere: A = 4πr^2
- Volume of a sphere: V = (4/3)πr^3
- A great circle has the same center as the sphere and radius r.
- A plane that cuts a sphere but does not pass through the center forms a small circle.
Vocabulary
- Sphere
- A sphere is the set of all points in three-dimensional space that are a fixed distance from a center point.
- Radius
- The radius is a segment from the center of a sphere to any point on its surface.
- Diameter
- The diameter is a segment through the center of a sphere with endpoints on opposite sides of the sphere.
- Great Circle
- A great circle is the largest possible circle on a sphere, formed by slicing the sphere through its center.
- Hemisphere
- A hemisphere is one of two equal halves of a sphere formed by a great circle.
Common Mistakes to Avoid
- Calling every circle on a sphere a great circle is wrong because only circles whose planes pass through the sphere's center are great circles.
- Using diameter instead of radius in A = 4πr^2 or V = (4/3)πr^3 is wrong because both formulas require r, not d.
- Thinking a small circle divides a sphere into two equal halves is wrong because only a great circle creates two congruent hemispheres.
- Confusing surface area with volume is wrong because surface area measures the outside covering in square units, while volume measures the space inside in cubic units.
Practice Questions
- 1 A sphere has radius 6 cm. Find its diameter, surface area, and volume in terms of π.
- 2 A sphere has diameter 14 m. Find the circumference of a great circle on the sphere.
- 3 A plane cuts a sphere and forms a circle. Explain how you can tell whether the circle is a great circle or a small circle.