A rhombus is a parallelogram with all of its sides being equal. A square is a rhombus with all of the angles being equal as well as all of the sides. Both squares and rhombuses have perpendicular diagonal bisectors that split each diagonal into 2 equal pieces, and also splitting the quadrilateral into 4 equal right triangles.

With this being said, we know the Pythagorean Theorem would work great in this situation, using half of each diagonal as the two legs of the right triangle.

\(\displaystyle \left ( 8 cm \right )^{2}+\left ( 6cm \right )^{2}=h^{2}\) where \(\displaystyle h\) is the hypotenuse.

\(\displaystyle h^{2}=100\) \(\displaystyle cm^{2}\) -->  \(\displaystyle h=10cm\)

We find the hypotenuse to be 10cm. Since the hypotenuse of each triangle is a side of the rhombus, we have found what we need to find the perimeter.

Each side is the same, so we add all 4 sides to find the perimeter.

\(\displaystyle 10+10+10+10=40\ cm\)

We should begin with a picture.

We should recall several things.  First, all four sides of a rhombus are congruent, meaning that if we find one side, we can simply multiply by four to find the perimeter.  Second, the diagonals of a rhombus are perpendicular bisectors of each other, thus giving us four right triangles and splitting each diagonal in half.  We therefore have four congruent right triangles.  Using Pythagorean Theorem on any one of them will give us the length of our sides.

With a side length of 17, our perimeter is easy to obtain.

\(\displaystyle \small P=4\cdot17=68\)

Our perimeter is 68 units.

In order to solve this problem, use the given information to work backwards to find a side length of the rhombus:

\(\displaystyle A=(base\times altitude)\)
\(\displaystyle 42=(base \times 7)\)
\(\displaystyle base=\frac{42}{7}=6\)

Then apply the perimeter formula:

\(\displaystyle P=4S\), where \(\displaystyle S=\) a side of the rhombus. 

\(\displaystyle P=4(6)=24\) 

In order to solve this problem, use the given information to work backwards to find a side length of the rhombus: 

\(\displaystyle A=(base\times altitude)\)


\(\displaystyle 15=(base\times 2)\)


\(\displaystyle base=\frac{15}{2}=7.5\)

Since, perimeter \(\displaystyle =\) \(\displaystyle p=4(S)\), where \(\displaystyle S\) is equal to \(\displaystyle 7.5\)

Perimeter=\(\displaystyle 4\times7.5=30\)

To find the perimeter, apply the formula: \(\displaystyle p=4(S)\), where \(\displaystyle S=9.5\)

\(\displaystyle p=4(9.5)=38\)

In order to solve this problem, use the given information to work backwards to find a side length of the rhombus: 

\(\displaystyle A=(base\times altitude)\)


\(\displaystyle 66=(base\times 6)\)


\(\displaystyle base=\frac{66}{6}=11\)


\(\displaystyle P=4S\), where \(\displaystyle S=11\)

\(\displaystyle P=4(11)=44\)

To find the perimeter of this rhombus, apply the formula: \(\displaystyle p=4(S)\), where \(\displaystyle S=4\frac{3}{4}=\frac{19}{4}\)

\(\displaystyle P=4(\frac{19}{4})=19\)

In order to solve this problem, use the given information to work backwards to find a side length of the rhombus: 

\(\displaystyle A=(base\times altitude)\)


\(\displaystyle 120=(base\times 7.5)\)


\(\displaystyle base=120\div7.5=16\)

\(\displaystyle P=4S\), where\(\displaystyle S=16\)

\(\displaystyle P=4(16)=64\)

To find the perimeter, first convert \(\displaystyle \frac{5}{6}\) foot into the equivalent amount of inches. Since, \(\displaystyle \frac{5}{6}=\frac{10}{12}\) and \(\displaystyle \frac{12}{12}=1foot\), \(\displaystyle \frac{5}{6}\) is equal to \(\displaystyle 10\) inches. 

Then apply the formula \(\displaystyle P=4S\), where \(\displaystyle S\) is equal to the length of one side of the rhombus. 

Since, \(\displaystyle S=10\)

The solution is:

\(\displaystyle P=4(10)=40\)

In order to find the perimeter of this rhombus, first convert \(\displaystyle 6\frac{5}{8}\) from a mixed number to an improper fraction: 

\(\displaystyle 6\frac{5}{8}=\frac{48+5}{8}=\frac{53}{8}\)

Then apply the formula: \(\displaystyle p=4S\), where \(\displaystyle S\) is equal to one side of the rhombus. 

Since, \(\displaystyle S=\frac{53}{8}\) the solution is:

\(\displaystyle p=4(\frac{53}{8})\)