Calculating Sphere Volume from Surface Area

This problem asks us to find the volume of a sphere given its surface area. We need to use the formulas for the surface area and volume of a sphere to solve this.

Understanding Sphere Formulas

A sphere is a perfectly round geometrical object in three-dimensional space that is the surface of a perfectly round ball. The key properties related to this problem are its surface area and volume.

• The formula for the surface area of a sphere is \(A = 4 \pi r^2\), where \(A\) is the surface area and \(r\) is the radius of the sphere.
• The formula for the volume of a sphere is \(V = \frac{4}{3} \pi r^3\), where \(V\) is the volume and \(r\) is the radius of the sphere.

Step-by-Step Solution

We are given that the surface area of the sphere is \(64 \pi \) cm². We can use this information and the surface area formula to find the radius of the sphere.

Step 1: Find the Radius of the Sphere

We set the given surface area equal to the formula for the surface area:

\(4 \pi r^2 = 64 \pi \)

To find \(r^2\), we divide both sides by \(4 \pi \):

\(r^2 = \frac{64 \pi }{4 \pi }\)

\(r^2 = 16\)

Now, we take the square root of both sides to find the radius \(r\):

\(r = \sqrt{16}\)

\(r = 4\) cm

So, the radius of the sphere is 4 cm.

Step 2: Calculate the Volume of the Sphere

Now that we have the radius (\(r = 4\) cm), we can use the formula for the volume of a sphere:

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

Substitute the value of \(r\) into the formula:

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

Calculate \(4^3\):

\(4^3 = 4 \times 4 \times 4 = 16 \times 4 = 64\)

Substitute \(64\) back into the volume formula:

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

Multiply \(\frac{4}{3}\) by \(64\):

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

\(V = \frac{256}{3} \pi \)

The volume of the sphere is \(\frac{256}{3} \pi \) cm³.

Summary of Steps

Comparing our calculated volume with the given options, we find that it matches option 4.

Revision Table: Sphere Geometry Formulas

Additional Information: Understanding Spheres

A sphere is a fundamental 3D shape. It is defined as the set of all points in three-dimensional space that are equally distant from a given point, called the center. The constant distance is called the radius.

• Unlike shapes like cubes or cylinders, a sphere has no edges or vertices.
• Every cross-section through the center of a sphere is a circle called a 'great circle'. The equator on Earth is an example of a great circle (if Earth were a perfect sphere).
• The formulas for surface area and volume of a sphere were first rigorously derived by the ancient Greek mathematician Archimedes. He showed that the surface area of a sphere is equal to the lateral surface area of a circumscribed cylinder, and the volume of a sphere is two-thirds the volume of a circumscribed cylinder.