The surface area of a sphere is proportional to the square of its radius. How many times larger is the surface area if the radius is a. doubled? b. tripled? c. halved (divided by 2 )? d. divided by 3 ?

For a sphere, the surface area formula is crucial to understand. The formula is given by: \[ A = 4 \pi r^2 \] where \(A\) represents the surface area and \(r\) is the radius of the sphere. This relationship highlights that the surface area of a sphere depends on the square of its radius. When the radius changes, the effect on the surface area is quite significant due to the squared term. Keep this in mind when we explore various proportional changes in the radius.

The concept of proportional relationships is vital in this exercise. Since the surface area \(A\) is proportional to the square of the radius \(r^2\), any change in the radius will result in a change in the surface area, according to this square factor. This means:

• If the radius is doubled (\(2r\)), the surface area will increase by a factor of \(2^2 = 4\).
• If the radius is tripled (\(3r\)), the surface area will increase by a factor of \(3^2 = 9\).
• If the radius is halved (\(\frac{r}{2}\)), the surface area will decrease by a factor of \((\frac{1}{2})^2 = \frac{1}{4}\).
• If the radius is divided by 3 (\(\frac{r}{3}\)), the surface area will decrease by a factor of \((\frac{1}{3})^2 = \frac{1}{9}\).

The proportional relationship makes it easier to predict how the surface area changes without recalculating from scratch each time.

Changing the radius of a sphere directly affects its surface area due to the proportional relationship to the square of the radius. Let's explore this with a few examples:

1. Doubling the Radius
When the radius is doubled from \(r\) to \(2r\): \[ A' = 4 \pi (2r)^2 = 16 \pi r^2 \] The surface area increases by 4 times.

2. Tripling the Radius
When the radius is tripled from \(r\) to \(3r\): \[ A' = 4 \pi (3r)^2 = 36 \pi r^2 \] The surface area increases by 9 times.

3. Halving the Radius
When the radius is halved from \(r\) to \(\frac{r}{2}\): \[ A' = 4 \pi \left( \frac{r}{2} \right)^2 = \pi r^2 \] The surface area decreases to one-quarter.

4. Dividing the Radius by Three
When the radius is divided by three from \(r\) to \(\frac{r}{3}\): \[ A' = 4 \pi \left( \frac{r}{3} \right)^2 = \frac{4 \pi r^2}{9} \] The surface area decreases to one-ninth.

Understanding how these changes affect the surface area can help visualize the importance of the radius in determining the size of a sphere's surface area.