Understanding Sphere Surface Area Calculation

The question asks us to find the new surface area of a sphere if its radius is doubled, given the original surface area. This involves understanding the formula for the surface area of a sphere and how it changes with the radius.

Sphere Surface Area Formula

The surface area (\(A\)) of a sphere with radius (\(r\)) is given by the formula:

\[A = 4 \pi r^2\]

Here, \(\pi\) is a mathematical constant approximately equal to 3.14159.

Analyzing the Effect of Doubling the Radius

Let the original radius of the sphere be \(r\). The original surface area is given as \(A = 5544 \text{ cm}^2\).

Now, the radius of the sphere is doubled. Let the new radius be \(r'\). According to the problem:

\[r' = 2r\]

We need to find the new surface area, let's call it \(A'\), for the sphere with radius \(r'\).

Using the surface area formula for the new sphere:

\[A' = 4 \pi (r')^2\]

Substitute the expression for \(r'\) in terms of \(r\):

\[A' = 4 \pi (2r)^2\]

Now, simplify the term \((2r)^2\):

\[(2r)^2 = 2^2 \times r^2 = 4r^2\]

Substitute this back into the formula for \(A'\):

\[A' = 4 \pi (4r^2)\]

Rearrange the terms to compare it with the original surface area formula:

\[A' = 4 \times (4 \pi r^2)\]

We know that the original surface area \(A = 4 \pi r^2\). So, we can substitute \(A\) into the equation for \(A'\):

\[A' = 4 \times A\]

This shows that if the radius of a sphere is doubled, its surface area becomes four times the original surface area.

Calculating the New Surface Area

We are given that the original surface area \(A = 5544 \text{ cm}^2\).

Using the relationship \(A' = 4 \times A\), we can calculate the new surface area:

\[A' = 4 \times 5544 \text{ cm}^2\]

Let's perform the multiplication:

• \(4 \times 5000 = 20000\)
• \(4 \times 500 = 2000\)
• \(4 \times 40 = 160\)
• \(4 \times 4 = 16\)

Adding these values:

\(20000 + 2000 + 160 + 16 = 22176\)

So, the new surface area \(A'\) is \(22176 \text{ cm}^2\).

Summary of Calculation

The surface area of the new sphere is \(22176 \text{ cm}^2\).

Revision Table: Key Sphere Formulas

Additional Information: Scaling of Geometric Shapes

This problem illustrates a general principle about how areas scale with changes in linear dimensions. For any two-dimensional shape, if all linear dimensions are scaled by a factor \(k\), the area is scaled by a factor of \(k^2\).

• In this case, the linear dimension is the radius, which was scaled by a factor \(k=2\) (doubled).
• The surface area is a two-dimensional measure (it's an area on the 3D surface).
• Therefore, the surface area is scaled by a factor of \(k^2 = 2^2 = 4\).

Similarly, for a three-dimensional shape (like the volume of a sphere), if all linear dimensions are scaled by a factor \(k\), the volume is scaled by a factor of \(k^3\).

• If the radius of a sphere is doubled (\(k=2\)), its volume will become \(2^3 = 8\) times the original volume.

Understanding this scaling principle can help solve many geometry problems quickly.