Sphere: Volume, Surface Area, and Great Circle Explained
Sphere: Volume, Surface Area, and Great Circle Explained
TL;DR
For a sphere of radius r, the volume is ( \frac{4}{3}\pi r^3 ) and the surface area is ( 4\pi r^2 ) - exactly four times the area of its great circle (( \pi r^2 )). Every flat slice through a sphere is a circle, and the biggest such slice, through the centre, is the great circle.
Last updated on July 21, 2022
9 min read
The Mensuration Of A Sphere
This article is the measurement deep-dive for the sphere: its volume, its surface area, its great circle, and its cross-sections. In short, a sphere is the set of all points in three-dimensional space that sit the same distance (the radius r) from a fixed centre point.
The two formulas, and where they come from
Volume. [ V = \frac{4}{3}\pi r^3 ] Here r is the radius and ( \pi \approx 3.14159 ). This formula originates from considering the sphere's surface as made of tiny pyramids pointing to the center.
- Surface area. [ S = 4\pi r^2 ] The surface area of a sphere is four times the area of its great circle. The equations below summarize key measurements:
| Quantity | Formula | Notes |
|---|---|---|
| Volume | ( V = \frac{4}{3}\pi r^3 ) | grows with the cube of r |
| Surface area | ( S = 4\pi r^2 ) | four great circles |
| Great circle area | ( \pi r^2 ) | largest cross-section |
| Great circle circumference | ( 2\pi r ) | the sphere's "equator" |
| Diameter | ( d = 2r ) | widest straight line through it |
The great circle and cross-sections
- A cut through the centre gives the great circle. Every great circle splits the sphere into two equal hemispheres.
- A cut that misses the centre gives a smaller circle.
Great circles matter beyond geometry: the shortest path between two points on a globe follows a great circle.
Examples Of Sphere Mensuration
Example 1
Find the volume of a sphere with radius 6 cm. Use ( \pi \approx 3.14 ). [ V = \frac{4}{3}\pi r^3 = \frac{4}{3} \times 3.14 \times 6^3 = 904.32 , \text{cm}^3 ]
Example 2
Find the surface area of a sphere with diameter 10 cm. The correct method is to halve the diameter first: ( r = \frac{10}{2} = 5 , \text{cm} ). [ S = 4\pi r^2 = 314 , \text{cm}^2 ]
Example 3
A sphere has surface area 4( \pi r^2 = 616 , \text{cm}^2 ). Find its radius. [ 4 \times \frac{22}{7} \times r^2 = 616 \Rightarrow r = 7 , \text{cm} ]
Example 4
Find the volume of a hemisphere with radius 3 cm. [ V_{\text{hemisphere}} = \frac{1}{2} \times \frac{4}{3}\pi r^3 = 56.52 , \text{cm}^3 ]
Example 5
A spherical balloon's radius doubles from 5 cm to 10 cm. By what factor does its volume grow? [ \frac{V_2}{V_1} = \frac{1000}{125} = 8 ]
Example 6
The great circle of a sphere has circumference 44 cm. Find the sphere's radius. [ r = 7 , \text{cm} ]
Why The Sphere's Formulas Matter
The sphere is nature's default shape for enclosing the most volume with the least surface area. Its implications can be seen in biology and engineering.
Mistakes To Watch For With Sphere Formulas
Mistake 1: Using the diameter where the radius belongs
Halve the diameter first, then substitute.
Mistake 2: Confusing the volume and surface-area formulas
Anchor them by units: Volume is cubic; Surface area is square.
Mistake 3: Forgetting the flat face when measuring a hemisphere's surface
The total surface area of a solid hemisphere is the curved surface plus the flat base.
Key Takeaways
- A sphere's volume is ( \frac{4}{3}\pi r^3 ) and surface area is ( 4\pi r^2 ).
- The surface area equals four times the great-circle area.
- Doubling the radius multiplies volume by 8 and surface area by 4.