Geometry - How to find the perimeter of a rhombus

A rhombus has a side length of $\frac{5}{6}$ foot, what is the length of the perimeter (in inches).

inches

feet

inches

Explanation

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

Then apply the formula , where is equal to the length of one side of the rhombus.

Since,

The solution is:

Find the perimeter of a rhombus if it has an area of and a diagonal length of .

Explanation

Recall the following properties of a rhombus, as shown by the figure above: The diagonals of a rhombus are perpendicular and they bisect each other. Thus, if we know the lengths of the diagonals, we can use the Pythagorean theorem to find the length of a side of the rhombus.

Recall how to find the area of a rhombus:

$\text{Area}=\frac{(\text{diagonal 1})(\text{diagonal 2})}{2}$

Since we are given the length of one diagonal and the area, we can find the length of the second diagonal.

$\text{diagonal 2}=\frac{2(\text{Area})}{\text{diagonal 1}}$

Plug in the given values to find the length of the second diagonal.

$\text{diagonal 2}=\frac{2(1200)}{40}=60$

Now, notice that the halves of each diagonal make up a right triangle that has the side length of the rhombus as its hypotenuse.

$\text{Half diagonal 1}=\frac{40}{2}=20$

$\text{Half diagonal 2}=\frac{60}{2}=30$

Now, use these half values in the Pythagorean Theorem to find the length of the side of a rhombus.

$\text{Side of Rhombus}=\sqrt{20^2+30^2}=\sqrt{1300}$

Finally, recall that a rhombus has four equal side lengths. To find the perimeter, multiply the length of a side by four.

$\text{Perimeter of rhombus}=4\sqrt{1300}=144.22$

Make sure to round to places after the decimal.

Find the perimeter of a rhombus if it has diagonals of the following lengths: and

Explanation

Recall that the diagonals of a rhombus bisect each other and are perpendicular to one another.

In order to find the length of the side, we can use Pythagorean's theorem. The lengths of half of the diagonals are the legs, and the length of the side of the rhombus is the hypotenuse.

First, find the lengths of half of the diagonals.

$\frac{\text{Diagonal 1}}{2}=\frac{8}{2}=4$

$\frac{\text{Diagonal 2}}{2}=\frac{12}{2}=6$

Next, substitute these half diagonals as the legs in a right triangle using the Pythagorean theorem.

$4^2+6^2=\text{side length}^2$

$\text{side length}^2=16+36=52$

$\text{side length}=2\sqrt{13}$

Since the sides of a rhombus all have the same length, multiply the side length by in order to find the perimeter.

$\text{Perimeter}=4(\text{side length})$

$\text{Perimeter}=4(2\sqrt{13})$

Solve and round to two decimal places.

$\text{Perimeter}=28.84$

Find the perimeter of a rhombus if it has an area of and a diagonal of .

Explanation

Recall how to find the area of a rhombus:

$\text{Area}=\frac{(\text{diagonal 1})(\text{diagonal 2})}{2}$

Since we are given the length of one diagonal and the area, we can find the length of the second diagonal.

$\text{diagonal 2}=\frac{2(\text{Area})}{\text{diagonal 1}}$

Plug in the given values to find the length of the second diagonal.

$\text{diagonal 2}=\frac{2(180)}{12}=30$

Now, notice that the halves of each diagonal make up a right triangle that has the side length of the rhombus as its hypotenuse.

$\text{Half diagonal 1}=\frac{12}{2}=6$

$\text{Half diagonal 2}=\frac{30}{2}=15$

Now, use these half values in the Pythagorean Theorem to find the length of the side of a rhombus.

$\text{Side of Rhombus}=\sqrt{6^2+15^2}=\sqrt{261}$

Finally, recall that a rhombus has four equal side lengths. To find the perimeter, multiply the length of a side by four.

$\text{Perimeter of rhombus}=4\sqrt{261}=64.62$

Make sure to round to places after the decimal.

Find the perimeter of a rhombus that has a side length of $6\frac{5}{8}$ .

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

$p=\frac{53}{8}$

$p=\frac{1}{2}$

$p=\frac{8}{53}$

$p=8\div53$

Explanation

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

$6\frac{5}{8}=\frac{48+5}{8}=\frac{53}{8}$

Then apply the formula: , where is equal to one side of the rhombus.

Since, $S=\frac{53}{8}$ the solution is:

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

A rhombus has an area of square units and an altitude of . Find the perimeter of the rhombus.

Explanation

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

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

Then apply the perimeter formula:

, where a side of the rhombus.

Find the perimeter of a rhombus if it has an area of and a diagonal of .

Explanation

Recall how to find the area of a rhombus:

$\text{Area}=\frac{(\text{diagonal 1})(\text{diagonal 2})}{2}$

Since we are given the length of one diagonal and the area, we can find the length of the second diagonal.

$\text{diagonal 2}=\frac{2(\text{Area})}{\text{diagonal 1}}$

Plug in the given values to find the length of the second diagonal.

$\text{diagonal 2}=\frac{2(64)}{16}=8$

Now, notice that the halves of each diagonal make up a right triangle that has the side length of the rhombus as its hypotenuse.

$\text{Half diagonal 1}=\frac{16}{2}=8$

$\text{Half diagonal 2}=\frac{8}{2}=4$

Now, use these half values in the Pythagorean Theorem to find the length of the side of a rhombus.

$\text{Side of Rhombus}=\sqrt{8^2+4^2}=\sqrt{80}$

Finally, recall that a rhombus has four equal side lengths. To find the perimeter, multiply the length of a side by four.

$\text{Perimeter of rhombus}=4\sqrt{80}=35.77$

Make sure to round to places after the decimal.

Find the perimeter of a rhombus if it has diagonals of the following lengths: and .

Explanation

Recall that the diagonals of a rhombus bisect each other and are perpendicular to one another.

In order to find the length of the side, we can use Pythagorean's theorem. The lengths of half of the diagonals are the legs, and the length of the side of the rhombus is the hypotenuse.

First, find the lengths of half of the diagonals.

$\frac{\text{Diagonal 1}}{2}=\frac{14}{2}=7$