The volume of a right prism is calculated by multiplying the area of the base by the height. So, for a square prism of side length aa and height hh the volume is a×a×h=a2ha×a×h=a2h.

**Volume of Prisms**

Calculate the area of the base and multiply by the height to get the volume of a prism.

Now, what happens to the surface area if one dimension is multiplied by a constant? For example, how does the surface area change when the height of a rectangular prism is divided by 2?

The size of a prism is specified by the length of its sides. The prism in the diagram has sides of lengths LL, bb and hh.

- Consider enlarging all sides of the prism by a constant factor xx, where x>1x>1. Calculate the volume and surface area of the enlarged prism as a function of the factor xx and the volume of the original volume.
- In the same way as above now consider the case, where 0<x<10<x<1. Now calculate the reduction factor in the volume and the surface area.

- Step 1.
** Identify : **
The volume of a prism is given by:
V=L×b×hV=L×b×h

The surface area of the prism is given by:
A=2×(L×b+L×h+b×h)A=2×(L×b+L×h+b×h)

- Step 2.
** Rescale : **
If all the sides of the prism get rescaled, the new sides will be:

L
'
=
x
×
L
b
'
=
x
×
b
h
'
=
x
×
h
L
'
=
x
×
L
b
'
=
x
×
b
h
'
=
x
×
h

(1)
The new volume will then be given by:

V
'
=
L
'
×
b
'
×
h
'
=
x
×
L
×
x
×
b
×
x
×
h
=
x
3
×
L
×
b
×
h
=
x
3
×
V
V
'
=
L
'
×
b
'
×
h
'
=
x
×
L
×
x
×
b
×
x
×
h
=
x
3
×
L
×
b
×
h
=
x
3
×
V

(2)
The new surface area of the prism will be given by:

A
'
=
2
×
(
L
'
×
b
'
+
L
'
×
h
'
+
b
'
×
h
'
)
=
2
×
(
x
×
L
×
x
×
b
+
x
×
L
×
x
×
h
+
x
×
b
×
x
×
h
)
=
x
2
×
2
×
(
L
×
b
+
L
×
h
+
b
×
h
)
=
x
2
×
A
A
'
=
2
×
(
L
'
×
b
'
+
L
'
×
h
'
+
b
'
×
h
'
)
=
2
×
(
x
×
L
×
x
×
b
+
x
×
L
×
x
×
h
+
x
×
b
×
x
×
h
)
=
x
2
×
2
×
(
L
×
b
+
L
×
h
+
b
×
h
)
=
x
2
×
A

(3)
- Step 3.
** Interpreting the above results : **
- We found above that the new volume is given by:
V'=x3×VV'=x3×V
Since x>1x>1, the volume of the prism will be increased by a factor of x3x3.
The surface area of the rescaled prism was given by:
A'=x2×AA'=x2×A
Again, since x>1x>1, the surface area will be increased by a factor of x2x2. Surface areas which are two dimensional increase with the square of the factor while volumes, which are three dimensional, increase with the cube of the factor.
- The answer here is based on the same ideas as above.
In analogy, since here 0<x<10<x<1, the volume will be reduced by a factor of x3x3 and the surface area will be decreased by a factor of x2x2

When the length of one of the sides is multiplied by a constant the effect is to multiply the original volume by that constant, as for the example in Figure 8.

Comments:"Accessible versions of this collection are available at Bookshare. DAISY and BRF provided."