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 𝑟.

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

  • A27
  • B149
  • C134
  • D9
  • E3

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

  • A9
  • B27
  • C3
  • D134
  • E149

Q2:

Consider a cube that initially has side length 𝑥.

If 𝑥 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 𝑥.

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

  • A𝑉=𝑥
  • B𝑉=𝑥
  • C𝑉=𝑥
  • D𝑉=6𝑥
  • E𝑉=6𝑥

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

  • A𝐴=𝑥
  • B𝐴=6𝑥
  • C𝐴=𝑥
  • D𝐴=6𝑥
  • E𝐴=8𝑥

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

  • A𝑉𝐴=16𝑥
  • B𝑉𝐴=12𝑥
  • C𝑉𝐴=14𝑥
  • D𝑉𝐴=18𝑥
  • E𝑉𝐴=116𝑥

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, 𝑉, the surface area, 𝐴, and the volume-to-surface-area ratio, 𝑉𝐴, vary with the radius of a sphere.

Which line represents 𝑉?

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

Which line represents 𝐴?

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

Which line represents 𝑉𝐴?

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

Q6:

Consider a sphere that initially has a radius 𝑟.

If 𝑟 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 𝑉 of a sphere of radius 𝑟 is given by the formula 𝑉=43𝜋𝑟. The surface area 𝐴 of the same sphere is given by 𝐴=4𝜋𝑟. Which of the following equations correctly gives the ratio of the volume of the sphere to the surface area of the sphere, 𝑉𝐴?

  • A𝑉𝐴=13𝜋𝑟
  • B𝑉𝐴=13𝑟
  • C𝑉𝐴=13𝑟
  • D𝑉𝐴=14𝜋𝑟
  • E𝑉𝐴=13𝑟

Q9:

The graph shows how the volume, 𝑉, the surface area, 𝐴, and the ratio of the volume to the surface area, 𝑉𝐴, vary with side length for a cube.

Which line represents 𝑉?

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

Which line represents 𝐴?

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

Which line represents 𝑉𝐴?

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

Q10:

Consider a cube that initially has a side length of 𝑥.

If 𝑥 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

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