# 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