Surface area to volume ratio of prisms
Formulas for surface area and volume of a right rectangular prism
Remember that the surface area of a right rectangular prism is given in the table:
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The formula for surface area of a rectangular prism is
???A=2lw+2wh+2lh???
And the volume of a rectangular prism is the length times the width times the height.
???V=lwh???
Surface area to volume ratio
The ratio of the surface area, ???S???, to the volume, ???V???, can be expressed as a fraction ???S/V???, or converted to a decimal.
How to find the surface area of rectangular prisms
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Calculating the surface area to volume ratio of the right rectangular prism
Example
Calculate the surface area to volume ratio of a right rectangular prism that measures ???4\text{ cm}??? high, ???\text{6 cm}??? wide, and ???\text{8 cm}??? long. Express your answer as a decimal rounded to the nearest tenth.
Use ???A=2lw+2wh+2lh??? to find the surface area.
???S=2\cdot 8\cdot 6+2\cdot 6\cdot 4+2\cdot 8\cdot 4???
???S=208???
Use ???V=lwh??? to find the volume.
???V=8\cdot 6\cdot 4???
???V=192???
Now find the ratio of surface area to volume.
???\frac{S}{V}=\frac{208}{192}=\frac{13}{12}???
As a decimal rounded to the nearest tenth, the ratio is ???1.1???.
The ratio of the surface area, S, to the volume, V, can be expressed as a fraction S/V, or converted to a decimal.
Example
The surface area to volume ratio of a right rectangular prism is ???11/20???. The volume of the prism is ???1,140\text{ feet}^3???. What is the prism’s surface area?
We know
???\frac{S}{V}=\frac{11}{20}???
and that the volume is ???V=1,440\text{ feet}^3???. The surface area is
???\frac{S}{1,140}=\frac{11}{20}???
???\frac{S}{1,140}(1,140)=\frac{11}{20}(1,140)???
???S=627\ \text{feet}^2???
