#### Answer

The area of the rectangle is $48$ square centimeters.

#### Work Step by Step

To find the area of the rectangle, we need to know the dimensions of the rectangle and plug them into the following formula to calculate area:
$A = lw$, where $A$ is the area of the rectangle, $l$ is the length of the rectangle, and $w$ is the width of the rectangle.
Let us set up some expressions for this problem.
Let $w = width$.
If the length $l$ is seven more than three times the width, then we can express $l$ as follows:
$l = 3w + 7$
We are also given that the perimeter is $38$ cm. Let us use the formula to calculate the perimeter of a rectangle to figure out its dimensions. The perimeter of a rectangle is given by the following formula:
$P = 2l + 2w$, where $P$ is the perimeter, $l$ is the length, and $w$ is the width of the rectangle.
Let's plug in what we know into this formula:
$38 = 2(3w + 7) + 2w$
Let's use the distributive property first:
$38 = 6w + 14 + 2w$
Combine like terms on the right side of the equation:
$38 = 8w + 14$
Subtract $14$ from each side of the equation to isolate constants on one side of the equation:
$8w = 24$
Divide each side by $8$ to solve for $w$:
$w = 3$
Now that we have the value for $w$, we can substitute it into the expression for $l$ to find the length of the rectangle:
$l = 3w + 7$
Substitute $3$ for $w$:
$l = 3(3) + 7$
Multiply first, according to order of operations:
$l = 9 + 7$
Add to solve for $l$:
$l = 16$
We now have the dimensions of the rectangle, so we can substitute these values into the formula to find the area of the rectangle:
$A = lw$
Substitute the values for $l$ and $w$ in the formula:
$A = 3(16)$
Multiply to solve for $A$:
$A = 48$
The area of the rectangle is $48$ square centimeters.