# Worksheet: Surface Area to Volume Ratios

In this worksheet, we will practice finding the ratio between the surface area and volume of different shapes.

Q1:

Consider a sphere that initially has a radius .

If changes by a factor of 3, by what factor does the volume of the sphere change?

• A27
• B
• C
• D9
• E3

By what factor does the surface area of the sphere change?

• A9
• B27
• C3
• D
• E

Q2:

Consider a cube that initially has side length .

If increases by a factor of two, by what factor does the volume of the cube change?

By what factor does the surface area of the cube change?

Q3:

Consider a cube of side length .

Which of the following is the correct equation for the volume, , of the cube?

• A
• B
• C
• D
• E

Which of the following is the correct equation for the surface area, , of the cube?

• A
• B
• C
• D
• E

Which of the following is the correct equation for the ratio of the volume of the cube to its surface area?

• A
• B
• C
• D
• E

Q4:

If the surface area of a sphere doubles, by what factor does its volume change? Give your answer to 3 significant figures.

Q5:

The graph shows how the volume, , the surface area, , and the volume-to-surface-area ratio, , vary with the radius of a sphere. Which line represents ?

• AThe blue line
• BThe purple line
• CThe red line

Which line represents ?

• AThe red line
• BThe purple line
• CThe blue line

Which line represents ?

• AThe red line
• BThe blue line
• CThe purple line

Q6:

Consider a sphere that initially has a radius .

If increases by a factor of two, by what factor does the volume of the sphere change?

By what factor does the surface area of the sphere change?

Q7:

If the volume of a sphere doubles, by what factor does its surface area change? Give your answer to 3 significant figures.

Q8:

The volume of a sphere of radius is given by the formula . The surface area of the same sphere is given by . Which of the following equations correctly gives the ratio of the volume of the sphere to the surface area of the sphere, ?

• A
• B
• C
• D
• E

Q9:

The graph shows how the volume, , the surface area, , and the ratio of the volume to the surface area, , vary with side length for a cube. Which line represents ?

• AThe red line
• BThe black line
• CThe blue line

Which line represents ?

• AThe red line
• BThe blue line
• CThe black line

Which line represents ?

• AThe blue line
• BThe red line
• CThe black line

Q10:

Consider a cube that initially has a side length of .

If changes by a factor of 3, by what factor does the volume of the cube change?

• A81
• B3
• C9
• D6
• E27

By what factor does the surface area of the cube change?

• A9
• B3
• C27
• D81
• E6