Finding the Volume and Surface Area of a Cube

Learning Outcomes

  • Find the volume and surface area of a cube


 

A cube is a rectangular solid whose length, width, and height are equal. See Volume and Surface Area of a Cube, below. Substituting, s for the length, width and height into the formulas for volume and surface area of a rectangular solid, we get:

V=LWHS=2LH+2LW+2WHV=s⋅s⋅sS=2s⋅s+2s⋅s+2s⋅sV=s3S=2s2+2s2+2s2S=6s2\begin{array}{ccccc}V=LWH\qquad & & & & S=2LH+2LW+2WH\qquad \\ V=s\cdot s\cdot s\qquad & & & & S=2s\cdot s+2s\cdot s+2s\cdot s\qquad \\ V={s}^{3}\qquad & & & & S=2{s}^{2}+2{s}^{2}+2{s}^{2}\qquad \\ & & & & S=6{s}^{2}\qquad \end{array}


So for a cube, the formulas for volume and surface area are
V=s3V={s}^{3}
and
S=6s2S=6{s}^{2}
.

Volume and Surface Area of a Cube

For any cube with sides of length
ss
,

An image of a cube is shown. Each side is labeled s. Beside this is Volume: V equals s cubed. Below that is Surface Area: S equals 6 times s squared.
 

example

A cube is
2.52.5
inches on each side. Find its 1. volume and 2. surface area.

Solution

Step 1 is the same for both 1. and 2., so we will show it just once.

Step 1. Read the problem. Draw the figure and

label it with the given information.		 .
1.
Step 2. Identify what you are looking for. the volume of the cube
Step 3. Name. Choose a variable to represent it. let V = volume
Step 4. Translate.

Write the appropriate formula.
V=s3V={s}^{3}
Step 5. Solve. Substitute and solve.
V=(2.5)3V={\left(2.5\right)}^{3}


V=15.625V=15.625
Step 6. Check: Check your work.
Step 7. Answer the question. The volume is
15.62515.625
cubic inches.
2.
Step 2. Identify what you are looking for. the surface area of the cube
Step 3. Name. Choose a variable to represent it. let S = surface area
Step 4. Translate.

Write the appropriate formula.
S=6s2S=6{s}^{2}
Step 5. Solve. Substitute and solve.
S=6⋅(2.5)2S=6\cdot {\left(2.5\right)}^{2}


S=37.5S=37.5
Step 6. Check: The check is left to you.
Step 7. Answer the question. The surface area is
37.537.5
square inches.
 

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example

A notepad cube measures
22
inches on each side. Find its 1. volume and 2. surface area.





 

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