91Ó°ÊÓ

The surface area of a sphere is an important concept in geometry and is defined as the total area that the surface of the sphere occupies. For spheres, calculating the surface area involves a specific formula derived from geometric principles. This formula is given as:

• \(4\pi r^2\), where \(r\) is the radius of the sphere.

To apply this formula, you need to determine the radius first, which can be done by inspecting the sphere's equation. Once the radius is known, simply square it, multiply by \(4\pi\), and you have your result!
In our example, using the formula with a radius of 10, we get:

• \(4\pi(10^2) = 400\pi\).

This result tells us that if the sphere were unfolded and laid flat, it would cover an area of \(400\pi\) square units.

The volume of a sphere gives us the capacity of the sphere, essentially measuring how much space is enclosed within its surface. The volume is computed using another fundamental formula:

• \(\frac{4}{3}\pi r^3\)

Here, \(r\) represents the radius.
To find the volume, calculate the cube of the radius, multiply by \(\pi\), and then scale the result by \(\frac{4}{3}\).
Using our radius of 10 from the given problem, we substitute into the formula:

• \(\frac{4}{3}\pi(10^3) = \frac{4000}{3}\pi\)

This equation tells us the sphere holds a volume of \(\frac{4000}{3}\pi\) cubic units, which quantifies how much space is contained inside.

The radius of a sphere is a crucial measurement as it directly affects both the surface area and volume calculations. From geometry, the radius is the distance from the center of the sphere to any point on its surface. This parameter forms the backbone of many calculations and determines the sphere's overall size.
In mathematical equations like the one given in the problem, the radius is identified from the equation of a sphere:

• \((x - h)^2 + (y - k)^2 + (z - l)^2 = r^2\)

where \(r\) is the radius.
In many cases, especially when the sphere is centered at the origin like our problem, the equation simplifies to \(x^2 + y^2 + z^2 = r^2\).
Finding the radius is often the first step before proceeding to calculate other properties like surface area and volume. In our example, \(r^2 = 100\), thus \(r = 10\), a pivotal step for all subsequent calculations.