In geometry, a sphere is a solid that has a completely round shape defined in three-dimensional space. Mathematically, the sphere is defined as the set of points that are located at a constant distance from a fixed point in three-dimensional space. This constant distance is called the radius and the fixed point is called the center of the sphere. An example of a sphere in real life is a ball.

Here, we will learn about the main characteristics of these geometric figures. Also, we will learn about its most used formulas and we will apply them to solve some problems.

## Definition of a sphere

A sphere is a round geometric figure defined in three-dimensional space. The sphere is a three-dimensional solid, so it has a volume and a surface area. Similar to a circle, each point on the sphere is located the same distance from the center.

The shape of a sphere is round and has no faces. The sphere is a three-dimensional geometric figure that has a curved surface. Unlike other solids like the cube, the prism, the pyramid, a sphere does not have any flat surface. The spheres do not have vertices or edges either.

## Fundamental characteristics of a sphere

The following are the fundamental characteristics of spheres:

- A sphere is perfectly symmetrical.
- Spheres are not polyhedra.
- All points on the surface of the sphere are equidistant from the center.
- Spheres do not have faces, vertices, or edges.
- The spheres have a constant mean curvature.
- They have a constant width and circumference.

## Important sphere formulas

Spheres are three-dimensional figures, so their most important formulas are the volume formula and the surface area formula.

### Formula for the volume of a sphere

The volume of a sphere is calculated using the length of its radius. For this, we can use the following formula:

$latex V=\frac{4}{3}\pi {{r}^3}$ |

where *r* represents the length of the radius of the sphere.

### Formula for the surface area of a sphere

The surface area is equal to the entire outer surface of the sphere. To calculate this area, we also use the length of the radius and the following formula:

$latex A_{s}=4\pi {{r}^2}$ |

Start now: Explore our additional Mathematics resources

## Examples of sphere problems

The formulas seen above are used to solve the following exercises.

### EXAMPLE 1

What is the volume of a sphere that has a radius of 4 m?

**Solution:** We use the volume formula with the value $latex r=4$:

$latex V=\frac{4}{3}\pi {{r}^3}$

$latex V=\frac{4}{3}\pi {{(4)}^3}$

$latex V=\frac{4}{3}\pi (64)$

$latex V=268.1$

The volume is 268.1 m³.

### EXAMPLE 2

If a sphere has a radius of 7 m, what is its volume?

**Solution:** We use the volume formula with $latex r=7$:

$latex V=\frac{4}{3}\pi {{r}^3}$

$latex V=\frac{4}{3}\pi {{(7)}^3}$

$latex V=\frac{4}{3}\pi (343)$

$latex V=1436.8$

The volume is 1436.8 m³.

### EXAMPLE 3

A sphere has a radius of 3 m. What is its surface area?

**Solution:** We have the radius $latex r=3$, so we use this value in the formula for surface area:

$latex A_{s}=4\pi {{r}^2}$

$latex A_{s}=4\pi {{(3)}^2}$

$latex A_{s}=4\pi (9)$

$latex A_{s}=113.1$

The surface area is 113.1 m².

### EXAMPLE 4

What is the surface area of a sphere that has a radius of 6 m?

**Solution:** We use the value $latex r = 6$ in the formula for surface area:

$latex A_{s}=4\pi {{r}^2}$

$latex A_{s}=4\pi {{(6)}^2}$

$latex A_{s}=4\pi (36)$

$latex A_{s}=452.4$

The surface area is 452.4 m².

## See also

Interested in learning more about spheres? Take a look at these pages:

### Learn mathematics with our additional resources in different topics

**LEARN MORE**