# Sphere: Volume, Surface Area, and Great Circle Explained

[Geometry](/content/tag/geometry/index.html)

**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](/content/math/geometry/hemisphere/index.html).
- 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.